Contact integral geometry and the Heisenberg algebra
Dmitry Faifman
math.DG math.MG math.SG 53B15, 53D10, 52A39, 52A22

Formulae for the  Conjugate and the Subdifferential  of the  Supremum Function
    Pedro Pérez-Aros
    Received: date / Accepted: date
==============================================================================





Generalizing Weyl's tube formula and building on Chern's work, Alesker reinterpreted the Lipschitz-Killing curvature integrals as a family of valuations (finitely-additive measures with good analytic properties), attached canonically to any Riemannian manifold, which is universal with respect to isometric embeddings. In this note, we uncover a similar structure for contact manifolds. Namely, we show that a contact manifold admits a canonical family of generalized valuations, which are universal under contact embeddings. Those valuations assign numerical invariants to even-dimensional submanifolds, which in a certain sense measure the curvature at points of tangency to the contact structure. 
Moreover, these valuations generalize to the class of manifolds equipped with the structure of a Heisenberg algebra on their cotangent bundle. Pursuing the analogy with Euclidean integral geometry, we construct symplectic-invariant distributions on Grassmannians to produce Crofton formulas on the contact sphere. Using closely related distributions, we obtain Crofton formulas also in the linear symplectic space.






§ INTRODUCTION




 §.§ Background and motivation


Intrinsic volumes first appeared in convex geometry through Steiner's formula: given a compact convex body K⊂^n, (K+ϵ B^n)=∑_k=0^n ω_n-kμ_k(K) ϵ^n-k where B^j is unit Euclidean ball in ^j and ω_j is its volume. The coefficient μ_k(K), the k-th intrinsic volume, can be written explicitly for smooth K as μ_k(K)=c_n,k∫_∂ Kσ_n-1-k(κ_1,…,κ_n-1)dArea_K, where κ_j are the principal curvatures of ∂ K.
Alternatively, μ_k(K) can be given by integral-geometric Crofton formulas: μ_k(K)=c'_n,k∫__n-k(^n)χ(K ∩ E)dE where dE is the rigid motion invariant measure on the affine Grassmannian, and χ is the Euler characteristic. A third, axiomatic definition, was given by Hadwiger, who described the intrinsic volumes as the unique rigid-motion invariant continuous finitely additive measures on compact convex sets.

A closely related famous result is Weyl's tube formula <cit.>. It asserts that the volume of an ϵ-tube around a Riemannian manifold M embedded isometrically in Euclidean space ^N is a polynomial in ϵ≪1, whose coefficients are, remarkably, intrinsic invariants of the Riemannian manifold M, independent of the isometric embedding. These coefficients, now known as the intrinsic volumes of M, are intimately linked with the asymptotic expansion of the heat kernel, see <cit.>.

These results fall naturally in the domain of valuations on manifolds, a fairly young branch of valuation theory introduced by Alesker et al. in a sequence of works <cit.>, see also <cit.> for a survey. Valuation theory itself is a mixture of convex and integral geometry, originating in the early 20th century in works of Steiner, Blaschke, Chern and Santalo, as well as in Dehn's solution to Hilbert's third problem. Generally speaking, valuations are finitely additive measures on some family of nice subsets. In this note, there is typically some analytic restriction on the nature of the valuation, such as smoothness or smoothness with singularities, and the subsets are manifolds with corners or differentiable polyhedra.

Building on results of Chern, Alesker noticed a natural extension of Weyl's theorem for valuations: restricting to M the intrinsic volumes of ^N, (considered as valuations), yields an intrinsically defined family of valuations on M, now known as the Lipschitz-Killing valuations. Weyl's intrinsic volumes of M are then the integrals of the corresponding valuations. In a recent work, Fu and Wannerer <cit.> characterized the Lipschitz-Killing valuations as the unique family of valuations attached canonically (in a sense made precise therein) to arbitrary Riemannian manifolds that are universal to isometric embeddings.

Other spaces whose (smooth) valuation theories were considered in recent years include complex space forms <cit.>, the quaternionic plane <cit.>, the octonionic plane <cit.> and exceptional spheres <cit.>.

Numerous intriguing connections between convex and symplectic geometries are known to exist. To name a few: Viterbo's conjectured isoperimetric inequality for capacities of convex bodies <cit.>, was later shown by Artstein-Avidan, Karasev and Ostrover to imply Mahler's famous conjecture <cit.>. Capacities have been successfully studied up to a bounded factor using convex techniques, see <cit.> and <cit.>.
In <cit.>, links are established between systolic geometry, contact geometry, Mahler's conjecture and the geometry of numbers. Schäffer's dual girth conjecture for normed spaces has been proved using symplectic techniques <cit.>, and generalized further in <cit.> using hamiltonian group actions. In very recent works of Abbondandolo et al. <cit.>, some links are established between the geometry of the group of symplectomorphisms and systolic geometry of the 2-sphere. For an exposition of some of those connections, see <cit.>. 

The main objective for this work is to further explore the convex-symplectic link by studying the valuation theory of contact manifolds, using the Riemannian case and Weyl's principle as guides. 


 §.§ Informal summary

 We find that like in the Riemannian setting, contact manifolds possess a canonical family of valuations associated to them. We describe those valuations in two ways: geometrically through a curvature-type formula, and also dynamically through the invariants of a certain vector field at its singular points.

The contact valuations satisfy Weyl's principle of universality under embeddings, similarly to the valuation extension of the Weyl principle. Let us emphasize the role played by valuations in this phenomenon: It so happens that contact valuations only assume non-zero values on even dimensional submanifolds, while they live on odd-dimensional contact manifolds. Thus unlike the Riemannian case, Weyl's principle in the contact setting is only manifested in its extended to valuations form, as the statement in the original form becomes vacuous: all integrals of the contact valuations vanish.

Contact valuations are in fact an instance of a natural collection of valuations associated to the larger class of manifolds whose cotangent spaces admit a smoothly varying structure of Heisenberg algebras. The Heisenberg algebra provides a unifying link between contact, symplectic and metric geometries.

This path leads us to consider the valuation theory of the dual Heisenberg algebra, invariant under the group of the automorphisms of the Heisenberg algebra, which is closely related to the symplectic group. From this perspective, this is another step in the study of the valuation theory of non-compact Lie groups, which up to now has only been considered for the indefinite orthogonal group in <cit.>, and in a somewhat different framework for the special linear group, see <cit.>.

Further similarity to the metric setting is exhibited by the contact sphere, where we prove Crofton formulas and a Hadwiger-type theorem, thus establishing an integral-geometric and an axiomatic description of the contact valuations. 

Finally in the last part we explore the valuation theory of linear symplectic spaces. We show that there are no non-trivial invariant valuations, but nevertheless one can write oriented Crofton formulas for the symplectic volume of submanifolds.




 §.§ Main results


Let us very briefly recall or indicate the relevant notions. For precise definitions see sections <ref> and <ref>.

A contact manifold M^2n+1 is given by a maximally non-integrable hyperplane distribution, namely a smooth field of tangent hyperplanes H⊂ TM s.t. locally one can find α∈Ω^1(M) with H=(α) and dα|_H a non-degenerate 2-form.

A smooth valuation ϕ on an orientable manifold M^n, denoted ϕ∈𝒱^∞(M), is a finitely additive measure on the compact differentiable polyhedra of M, denoted 𝒫(M), which has the form ϕ(X)=∫_Xμ+∫_N^*Xω for some forms μ∈Ω^n(M) and ω∈Ω^n-1(S^*M). Here S^*M is the cosphere bundle, and N^*X is the conormal cycle of X, which is just the conormal bundle when X is a manifold. Orientability is not essential, and is only assumed to simplify the exposition. 

There is a natural filtration 𝒲^∞_n(M)⊂…⊂𝒲^∞_0(M)=𝒱^∞(M). Very roughly speaking, 𝒲^∞_j(M) consists of valuations which are locally homogeneous of degree at least j, for example 𝒲^∞_n(M) are the smooth measures on M.

The generalized valuations 𝒱^-∞(M) are, roughly speaking, distributional valuations: we allow ω and μ to be currents rather than smooth forms. Generalized valuations can be naturally evaluated on sufficiently nice subsets X ∈𝒫(M). The filtration 𝒲_j^∞(M) on 𝒱^∞(M) extends to a filtration  𝒲_j^-∞(M) on 𝒱^-∞(M).


	To any contact manifold M^2n+1 with contact distribution H⊂ TM there are canonically associated, linearly independent generalized valuations ϕ^M_2k∈𝒱^-∞(M), 0≤ k≤ n. They have the following properties:
	
		
  * ϕ^M_0 is the Euler characteristic, and ϕ^M_2k∈𝒲_2k^-∞(M)∖𝒲_2k+1^-∞(M).
		
  * ϕ_2k^M can be naturally evaluated on submanifolds in generic position relative to the contact structure (see Definition <ref>). For a generic closed hypersurface F, 
		
		
    ϕ^M_2k(F)=∑_T_pF=H_pϕ^M_2k(F,p)

		where the local contact area ϕ^M_2k(F,p) only depends on the germ of F at p. It is described explicitly below in equations (<ref>) and (<ref>).
		
		
  * Universality to restriction under embedding: if i:N^2m+1→ M^2n+1 is a contact embedding, then for k≤ m one has i^*ϕ^M_2k=ϕ^N_2k.
		
  * For a 2k-dimensional submanifold in general position F, ϕ^M_2k(F)≥ 0, with equality if and only if there are no contact tangent points.
		
  * The space of generalized valuations on M invariant under all contactomorphisms of M is spanned by (ϕ^M_2k)_k=0^n.
	


The universality with respect to embeddings is sometimes referred to as the Weyl principle. Thus we recover a Weyl principle in the contact setting. 
	
	
	The local contact areas ϕ^M_2k(F,p) can be given explicitly in two different ways, through a geometric or a dynamical approach.
	
		
  * From a dynamical point of view, ϕ^M_2k(F,p) encodes the invariants of the linearized vector field B∈𝒳(F) representing the characteristic foliation. While there are many such vector fields, there is a distinguished choice, to linear order, at the critical points - choose an arbitrary contact form α near p and let B be given by dα|_F(B,∙)=α|_F.  One then has 
		
    ϕ^M_2k(F,p)= ∧^2n-2k d_pB/| d_pB|

		
		
  * From a geometric point of view, 
		
    ϕ^M_k(F,p)= 2n  k | (S-h)|^-1D(S-h[2n-k], J[k]).

		Here D denotes the mixed discriminant, 
		J=([    0 -I_n;  I_n    0 ]), S is the second fundamental form of F at p, and h the second fundamental form of the contact distribution at p (see Definition <ref>), both written with respect to a frame compatible with the contact structure in a natural way at p, see Definition <ref>.
		Thus they are reminiscent of (certain symmetric functions of) the principal radii of an embedded hypersurface in a Riemannian manifolds. This point of view is applicable in the wider setting of DH manifolds, as described below.
	
	
We may extend ϕ_2k to a non-negative lower semicontinuos functional on all 2k-dimensional submanifolds with boundary, denoted _2k(F), the contact area of F. We observe that _2k(F)=0 if and only if F can be made nowhere tangent to the contact distribution by an arbitrarily small perturbation.
In Riemannian or Hermitian manifolds, a fair amount of the valuation theory appears already in the corresponding flat space, which can be thought of as the tangent space to the given manifold. In the contact setting, it is no longer true: the tangent space of a contact manifold does not itself inherit a contact structure. 

The main observation guiding this paper is that every cotangent space of a contact manifold is canonically the Heisenberg Lie algebra. We are thus led to study the valuation theory of general manifolds with such structure.

	A manifold X equipped with a hyperplane distribution H (called horizontal) and a smooth field of nowhere-degenerate forms ω∈Γ^∞(X, ∧ ^2 H_x^*⊗ (T_xX/H_x)) will be called a dual Heisenberg (DH) manifold. 



	
The space of valuations naturally associated to such manifolds turns out to resemble somewhat the Lipschitz-Killing space of valuations in Riemannian geometry, in particular, they exhibit universality with respect to embeddings.

Theorem <ref> is then the contact instance of the following general result.

 
	To any DH manifold M^2n+1 with horizontal distribution H⊂ TM there are canonically associated, generalized valuations ϕ^M_k∈𝒲_k^-∞(M), 0≤ k≤ 2n. They have the following properties:

	
  * ϕ^M_0 is the Euler characteristic, and ϕ^M_2k∈𝒲_2k^-∞(M)∖𝒲_2k+1^-∞(M).
	
	
  * ϕ_k^M can be naturally evaluated on submanifolds in generic position with respect to the horizontal distribution. For a generic closed hypersurface F, 
	
	
    ϕ^M_k(F)=∑_T_pF=H_pϕ^M_k(F,p)

	where ϕ^M_k(F,p) only depends on the germ of F at p. 
	
  * If i:N^2m+1→ M^2n+1 is a DH embedding, then for k≤ 2m one has i^*ϕ^M_k=ϕ^N_k.


The local contact areas ϕ_k^M(F,p) are given by the same curvature-type formula (<ref>) as in the contact case.

As an intermediate step of independent interest, we obtain a Hadwiger-type theorem for the dual Heisenberg algebra itself. We denote by U=^2n+1 the dual of the Heisenberg Lie algebra 𝔥_2n+1, and by _H(U) its automorphism group.


It holds that ^-∞(U)^_H(U) consists of even valuations. For 0≤ k≤ n, ^-∞_2k+1(U)^_H(U)={0} while ^-∞_2k(U)^_H(U)=1. The corresponding Klain sections are zero-order distributions (that is, regular Borel measures).

We also consider _H^+(U), the connected component of the identity. We prove


For 0≤ k≤ n, ^-∞_2k+1(U)^_H^+(U)={0} while ^-∞_2k(U)^^+_H(U)=2.
In the latter space, the _H(U)-invariant valuations are complemented by a one-dimensional space of odd valuations.


We then consider the standard contact sphere S^2n+1, which we identify with the oriented projectivization ℙ_+(V) of a symplectic space V=(^2n+2,ω).  We construct a canonical distribution μ_ω∈ℳ^-∞(_2k(V))^(V). We obtain the following Crofton formulas, establishing further common grounds with the Riemannian setting. 


	Define for 0≤ i≤ n the generalized valuations ψ_2i on S^2n+1 given by the Crofton formula 
	
    ψ_2i:=∫__2n+2-2i(V)χ(∙∩ E)dμ_ω(E).
 
	Then for certain explicit constans c^n_ij one has
	
	
    ψ_2i=∑_j=i^n c^n_ijϕ_2j.



We also establish a Hadwiger-type theorem for the contact sphere.


Both (ϕ_2k)_k=0^n and (ψ_2k)_k=0^n are bases of 𝒱^-∞(S^2n+1)^(2n+2). 


Finally, we find that while symplectic space and manifolds do not possess interesting invariant valuations, one can nevertheless write certain integral geometric formulas for symplectic volumes of manifolds.
We construct a canonical distribution on the affine oriented Grassmannian μ_ω∈ℳ^-∞(^+_2k(^2n)) which is translation- and (2n)-invariant, and odd to orientation reversal. We prove the following Crofton formula on symplectic linear space.

 Let F^2k⊂^2n be a C^1 compact, oriented submanifold with boundary. Then
	
    ∫_F ω^k=(-1)^κn k2n  2k^-1(2n-1)^κ/2^κ+1∫_^+_2n-2k(^2n) I(E, F)dμ_ω(E)

	where κ=min(k, n-k) and I is the oriented intersection index.
	




 §.§ Plan of the paper


In section <ref> we introduce notation and present the basic geometric facts we will use. In section <ref> we recall the basics of valuation theory and prove some lemmas we will need. In section <ref> we construct the canonical valuations on general DH manifolds and establish their universality to embeddings, proving Theorem <ref>. We also explore some geometric properties of those valuations.
In section <ref> we classify the translation-invariant valuations of the dual Heisenberg algebra, proving Theorems <ref> and <ref>, and note their relation to gaussian curvature. Apart from its intrinsic interest, the linear classification is needed for the uniqueness statement in Theorem <ref>, as well as for Theorem <ref>. In section <ref> we specialize the DH valuations to contact manifolds and prove Theorem <ref>. In particular, we give the dynamical description of the contact valuations. In section <ref> we construct symplectic-invariant distributions on linear and affine Grassmannians in symplectic space, which are used in the subsequent two sections. In section <ref> we consider the standard contact sphere. We produce Crofton formulas for ϕ_2k that are invariant under (V), proving theorems <ref> and <ref>, and compute some examples explicitly. We also bound from below the contact valuations of a convex set. Finally in section <ref>, we study the integral geometry of linear symplectic space, proving Theorem <ref>.


 §.§ Acknowledgements

I am grateful to Semyon Alesker, Andreas Bernig, Yael Karshon, Leonid Polterovich, Egor Shelukhin and Gil Solanes for fruitful discussions and insightful suggestions. I am also indebted to Andreas and Semyon for helpful comments on the first draft.

This research was partially supported by an NSERC Discovery Grant, and conducted while the author was employed as Coxeter Assistant Professor at the University of Toronto.



§ PRELIMINARIES



 §.§ Notation

We use σ_1 to denote the unit (n)-invariant measure on various homogeneous spaces of the special orthogonal group.
We write _k(V) for the k-Grassmannian in V, _k^+ for the oriented Grassmannian, and ^(+) for the affine (oriented) Grassmannian. 𝒦(V) is the set of compact convex subsets of V.

The one-dimensional space of real-valued Lebesgue measures over V is denoted (V). For a manifold M, |ω_M| denotes the line bundle of densities, whose fiber over x∈ M is (T_xM). 
We will write M^tr for the translation-invariant elements of a module M over V.
For a group G⊂GL(V), G is the group generated by G and all translations in V.

We will write Ω_-∞(M) for the space of currents on M, since we typically consider them as generalized differential forms.

Throughout the note, J=([    0 -I_n;  I_n    0 ]) is the standard real form of √(-1).  




 §.§ The symplectic group action on the Grassmannian


This subsection is the symplectic version of the corresponding section in <cit.> where (p,q) is considered.

Let V=(^2n, ω) be a symplectic space. Let X^k_r(V), 0≤ r≤κ:=⌊min(k, 2n-k)/2⌋ be the orbits of _k(V) under the real symplectic group (V), where 
    X^k_r(V)={E∈_k(V): ω|_E=min(k, 2n-k)-2r}.
 
When no confusion can arise we write simply X^k_r.
In the oriented Grassmannian _k^+(^2n), the double covers X^k, +_r of X^k_r are orbits of (V) when 0≤ r≤κ-1. For r=κ, there are two possibilities: for even k, the double cover of X^k, +_κ  splits into two open orbits, denoted X^k_±, corresponding to the different orientations induced by ω on the subspace; for odd k, X^k, +_κ is a single orbit.

We use the same notation for the corresponding (V)-orbits in the (oriented) affine Grassmannian. 

We will need a simple lemma from linear algebra, which is a linearized version of Witt's theorem.
 Take E∈ X^k_r. Write E_0=E∩ E^ω, E_0=min(k,2n-k)-2r. Let π_0: E^*⊗ V/E→ E_0^*⊗ E_0^* denote the map π_0(T)(x,y)=ω(Tx,y). Then
	
    T_EX^k_r={T:E→ V/E : π_0(T)∈^2 E_0^*}

 
Clearly π_0 is onto. Denote by π_E: V^*⊗ V→ E^*⊗ (V/E) the natural projection. Recall that 𝔰𝔭(V)={T∈𝔤𝔩(V): ω(Tx, y)=ω(Ty, x)  ∀ x,y∈ V}.
We have to show that π_E𝔰𝔭(V)=π_0^-1^2E_0^*.

The inclusion π_E𝔰𝔭(V)⊂π_0^-1^2E_0^* is immediate. In the other direction, take T:E→ V/E s.t. π_0T∈^2E_0^*. We should lift T to T̃∈𝔰𝔭(V).

Choose any subspaces E' ⊂ E, E”⊂ E^ω with E=E_0 ⊕ E', E^ω=E_0 ⊕ E”.  

Denote κ=min(k,2n-k). Then E'=k-κ+2r and E”=2n-k-κ+2r, and they are both non-degenerate. Moreover, E' ⊕ E”⊂ V is a non-degenerate subspace of dimension 2n-2κ+4r, and E_0 is an isotropic subspace of the non-degenerate space W:=(E' ⊕ E”)^ω, and W=2κ-4r. That is, E_0⊂ W is a Lagrangian subspace. Fix a Lagrangian complementing space F⊂ W s.t. W=E_0 ⊕ F. Then V=E' ⊕ E”⊕ E_0 ⊕ F.  

Let T_1 ∈(E,V) be a lift of T such that T_1(E') ⊂ E”⊕ F, T_1(E_0) ⊂ E”⊕ F ⊕ E' and 

ω( π'(T_1x),e):=ω(Te,x),   ∀x ∈E_0, e ∈E',

where π'(T_1x) is the E'-component of T_1x.

Note that ω gives identifications E^*=E' ⊕ F, as well as (E ⊕ E”)^*=E' ⊕ E”⊕ F. We extend T_1 to a map T_2 ∈(E ⊕ E”,V) by requiring T_2(E”) ⊂ E' ⊕ F and 

ω(T_2e”,e):=ω(T_1e,e”),   ∀e” ∈E”, e ∈E.


Note that ω(T_2e_1,e_2)=0 for all e_1,e_2 ∈  E”. 

Finally, we extend T_2 to a map T' ∈(V,V) by requiring T'(F) ⊂ E' ⊕ E”⊕ F and  

ω(T'f,x)=ω(T_2x,f),   ∀x ∈E ⊕E”, f ∈F.

Again we have ω(T'f,f')=0 for f,f' ∈ F. Then T' ∈𝔰𝔭(V) lifts T as required.



	There is a natural identification of N_EX^k_r with ∧^2 E_0^*.



    N_EX^k_r   =T_E_k(V)/T_EX^k_r=E^*⊗ (V/E)/ π_0^-1^2E_0^* 
       =(E_0^*⊗ E_0^*)/^2E_0^*=∧^2 E_0^*




	Fix L∈ℙ_+(V), and denote Z_L={E∈_k(V): L⊂ E}. Then Z_L intersects X^k_r transversally for all r.

Take E∈ Z_L∩ X^k_r, denote E_0=E∩ E^ω. Let A:E→ V/E be an arbitrary linear map. We will find a representation A=A_1+A_2 with A_1∈ T_EZ_L (that is, A_1|_L=0) and A_2∈ T_E X^k_r.  Fix l∈ L and write Al=v+E.

Case 1: L⊄E_0. We set A_2|_L:=A|_L, A_2|_E_0=0.

 Case 2: L⊂ E_0 and ω(v, E_0)= 0. Decompose E_0=L⊕ F and set A_2(l)=v+E, A_2(f)=0 for all f∈ F.

Case 3: L⊂ E_0 and ω(L, AL)≠ 0. Since ω(l, v)≠ 0, we may decompose E_0=E_0∩ v^ω⊕ L. We then set A_2(l)=v+E and A_2(x)=0 for x∈ E_0∩ v^ω.

Case 4: L⊂ E_0, ω(v, E_0)≠ 0 and ω(L, AL)= 0. Then we decompose E_0=L⊕ F⊕(u) where L⊕ F=v^ω∩ E_0 and ω(u, v)=1. Choose w∈ V such that ω(w, l)=-1 and w∈ F^ω. This is possible since l^ω≠ F^ω. Then set A_2l=v+E, A_2f=0 for f∈ F, and A_2u=w+E.

In all cases, extend A_2 arbitrarily to E. Thus in all cases, by Lemma <ref>, A_2∈ T_E X^k_r, and A_1:=A-A_2∈ T_EZ_L.








§ VALUATION THEORY




 §.§ Valuations on manifolds

For a manifold X, we let ℙ_X:=ℙ_+(T^*X) denote the oriented projectivization of its cotangent bundle and π:ℙ_X→ X the projection. ℙ_X has a canonical contact structure. A form ω∈Ω(ℙ_X) that vanishes when restricted to the contact distribution is usually called vertical. However, we will have several different notions of verticality, so we will call such forms Legendrian.


We say that a form ω∈Ω^d(ℙ_X) has horizontal degree at least k, written _Hω≥ k, if ω(v_1,…, v_d) vanishes whenever d+1-k of the vectors v_j are vertical, that is tangent to the fiber of π:ℙ_X→ X. 
The following is a simple reformulation.


{ω∈Ω(ℙ_X): _Hω≥ k} is the ideal in Ω(ℙ_X) generated by π^*Ω^k(X).



Assume ω=∑π^*ω_j ∧η_j, where ω_j∈Ω^k(X), η_j∈Ω^d-k(ℙ_X). Clearly if v_1,…,v_d+1-k are vertical vectors then π^*ω_j∧η_j(v_1,…, v_d+1-k,…)=0.

For the opposite direction, let us choose a Riemannian structure on X. Then ℙ_X is the sphere bundle on X. Fix coordinates dx_j on T_xX and dξ_j on T_ξS_xX. 
Then π^*Ω^k(X) is spanned over C^∞(ℙ _X) by {∧_i∈ Iπ^* dx_i: |I|=k}. Assume _Hω≥ k and decompose ω=∑_I, J  f_IJπ^*dx_I∧ dξ_J. 

Assume a multi-index I appears in the sum  with |I|<k, say I=(i_1,…, i_l), l<k, with corresponding J=(j_1,…,j_d-l). Let e_1 ^H,…, e_n^H, e_1^V, …, e_n-1^V be dual to π^*dx_1, …, π^*dx_n, dξ_1,…, dξ_n-1. Then  d-l≥ d+1-k so by assumption
0=ω(e_i_1^H, …, e_i_l^H, e_j_1^V, …, e_j_d-l^V )=f_IJ(x,ξ), so f_IJ=0. 



Let M be a smooth manifold, which we assume oriented for simplicity of exposition, and refer the reader to <cit.> for the general case. Denote by 𝒫(M) the compact differentiable polyhedra of M. We remark that manifolds with corners are an example of differentiable polyhedra, and refer to <cit.> for the definition of differentiable polyhedra.
The smooth valuations 𝒱^∞(M) consist of functionals ϕ:𝒫(M)→ which can be presented in the form ϕ(X)=∫_Xμ+∫_N^*Xω for some forms μ∈Ω^n(M) and ω∈Ω^n-1(ℙ_M). Here N^*X is the conormal cycle of X. It consists of codirections ξ∈ℙ_+(T^*_xM) which are non-positive on velocity vectors γ̇(0)∈ T_xM of all curves γ(t)∈ X with x=γ(0). The Euler characteristic χ is an important example of a smooth valuation. The smooth valuations over open subsets of X constitute a soft sheaf over X, see <cit.>. We denote the compactly supported valuations by 𝒱_c^∞(M), and 𝒲_i,c^∞(M):=𝒱_c^∞(M)∩𝒲_i^∞(M). 
We remark that whenever a valuation is evaluated on a subset, the subset is assumed compact.
There is a natural integration functional ∫_M:𝒱^∞_c(M)→ which is essentially evaluation on M, but more precisely - on a sufficiently large compact set. Both 𝒱^∞(M) and 𝒱^∞_c(M) inherit natural topologies from the corresponding spaces of pairs of forms. 

There is a natural filtration 𝒲^∞_n(M)⊂…⊂𝒲^∞_0(M)=𝒱^∞(M), introduced by Alesker <cit.>. We will use an equivalent description, which is the content of <cit.>. 


	𝒲^∞_k(M) consists of those valuations that can be represented by a pair (ω, μ) with _H ω≥ k.

In particular, 𝒲_n^∞(M) are just the smooth measures on M, denoted ℳ^∞(M).

Alesker defined a product structure 𝒲_i^∞(M)⊗𝒲_j^∞(M)→𝒲_i+j^∞(M) which turns 𝒱^∞(M) into a filtered algebra, whose unit is the Euler characteristic. It induces the following Alesker-Poincaré duality.
[Alesker <cit.>]
	The pairing 𝒲^∞_i(M)⊗𝒲_n-i,c^∞(M)→ given by
	(ϕ,ψ)↦ (ϕ·ψ)(M) is non-degenerate.


The presentation of ϕ∈𝒱^∞(M) by a pair of forms is not unique. There is an alternative faithful description due to Bernig and Bröcker <cit.>. In the following, a:ℙ_M→ℙ_M is the antipodal map in every fiber, and D is the Rumin differential introduced in <cit.>. We recall that Dω is the unique Legendrian form d(ω+η), where η∈Ω^n-1(ℙ_M) ranges over all Legendrian forms.

	If ϕ is represented by the pair (ω, μ)∈Ω^n-1(ℙ_M)×Ω^n(M), then 
    (T,C):=(a^*(Dω+π^*μ), π_*ω)∈Ω^n(ℙ_M)× C^∞(M)
 is determined by ϕ. They satisfy the relations dT=0, π_*T=(-1)^ndC, and T is Legendrian. Moreover, any (T,C) with those properties corresponds to a valuation.


We refer to (T,C) as the defining currents of ϕ (and often we refer just to T as the defining current). The reason for this terminology will become evident once we introduce generalized valuations. 

The Alesker-Poincaré duality is easy to describe using the defining currents.
[Bernig <cit.>]
	Let (ω, μ) represent ϕ_1, and let ϕ_2 have defining current (T_2, C_2). Then
	
    ∫_M ϕ_1·ϕ_2=∫_ℙ_Mω_1∧ T_2+∫_M C_2μ_1.



Let us describe the filtration on 𝒱^∞(M) through the defining currents. 

	Let (T, C)∈Ω^n(ℙ_M)× C^∞(X) be the defining current of ϕ∈𝒱^∞(M^n). 
	For 1≤ k≤ n, ϕ∈𝒲_k^∞(M^n) if and only if C=0 and _H T≥ k.

Consider first k=n. If ϕ∈𝒲_n^∞(M), we have C=0 and T=π^*μ for some μ∈Ω^n(M), which is clearly horizontal of degree at least n.
In the other direction, if _H T≥ n, it follows that T=f(ξ)π^*μ for some μ∈Ω^n(M) and f∈ C^∞ (ℙ_M). Now 0=dT=df∧π^*μ, that is df vanishes when restricted to the vertical fiber, so f=π^*f_1 for some f_1∈ C^∞(M). Hence T=π^*(f_1μ), so ϕ is just the measure f_1μ.

Assume now 1≤ k≤ n-1. Recall that by Alesker-Poincaré duality, ϕ∈𝒲_k^∞(M) if and only if for all ψ∈𝒲_n+1-k,c^∞(M), ∫_Mϕ·ψ=0. Combining Definition <ref> with eq. (<ref>), we conclude that ϕ∈𝒲_k^∞(M) if and only if for all ω∈Ω^n-1(ℙ_M) with _Hω≥ n+1-k, and all μ∈Ω^n(M), it holds that ∫_M(π_*(ω∧ T)+Cμ)=0. 

Assume that C=0 and _H T≥ k. Then by Lemma <ref>, ω∧ T=0 for all ω as above, and hence ϕ∈𝒲_k^∞(M).

In the other direction, assume ϕ∈𝒲_k^∞(M). Taking ω=0 and μ arbitrary we deduce C=0. It then follows that π_*(ω∧ T)=0, and since ω can have an arbitrarily small support, also that ω∧ T=0 whenever _Hω≥ n-k+1, so that _H T≥ k.



The valuations that appear naturally in contact manifolds are not smooth. To formally study them we will need the larger family of generalized valuations.


The generalized valuations  𝒱^-∞(M) are the continuous dual of 𝒱_c^∞(M), equipped with the weak topology.
  
A generalized valuation is uniquely determined by its defining current (T, C)∈ 𝒟_n-1(ℙ_M)×𝒟_n(M), see <cit.>. It can be an arbitrary pair of currents satisfying the three properties: T is Legendrian, π_*T=∂ C, and ∂ T=0. If ϕ_2∈𝒱^-∞(M) has defining current (T_2, C_2), it acts on smooth valuations ϕ_1∈𝒱_c^∞(M) represented by forms (ω_1, μ_1) through eq. (<ref>). We will write ϕ_2=[T_2, C_2], T_2=T(ϕ_2), C_2=C(ϕ_2).


	1. Given X∈𝒫(M), the evaluation at X functional χ_X:𝒱^∞(M)→ is a generalized valuation with defining currents C=[[X]] and T=[[N^*X]]. 
	
2. A generalized valuation ϕ_1 can be represented by a pair of generalized forms ω_1∈Ω_-∞^n-1(ℙ_M) and μ_1∈Ω_-∞^n(M). It acts on  ϕ_2=[T_2, C_2]∈𝒱^∞_c(M) through eq. (<ref>). It has defining currents given by T_1=a^*(Dω_1+π^*μ_1), C_1=π_*ω_1.


The filtration 𝒲_j^∞(M) on 𝒱^∞(M) extends to a filtration  𝒲_j^-∞(M) on 𝒱^-∞(M) by taking 𝒲_j^-∞(M) to be the annihilator of 𝒲_n+1-j, c^-∞(M).

We refer to <cit.> for the notion of the wavefront set of a distribution. Let us only record that for an oriented submanifold X⊂ M, ([[X]])=N^*X.

	The wavefront (ϕ) of ϕ∈𝒱^-∞(M) is the pair of wavefront sets ((T(ϕ)), (C(ϕ))). When (C(ϕ))=∅, we also write (ϕ)=(T(ϕ)).

Given a closed cone Γ⊂ T^*(ℙ_M)∖ 0, we let 𝒱^-∞_Γ(M):={ϕ∈𝒱^-∞(M):(T(ϕ))⊂Γ, (C(ϕ))=∅}. It inherits a topology from Hörmander's topology on the corresponding space of currents 𝒟_n-1, Γ(ℙ_M).

The Alesker-Poincaré duality extends to a pairing of generalized valuations, as long as the wavefront sets are in good relative position. We will only need the following weak version of Theorem 8.3 in <cit.>.

For a subset T∈ T^*ℙ_M, we write -T for its image under the antipodal map in every cotangent space T^*_x,ξ T^*(ℙ_M). Define also T^s:=T∪(-T)∪ aT∪ (-aT).


	Let Γ_1,Γ_2⊂ T^*(ℙ_M)∖ 0 be closed cones, Γ_j=Γ_j^s. Assume that Γ_1∩Γ_2=∅. Then there is a jointly sequentially continuous pairing
	
    𝒱^-∞_Γ_1(M)⊗𝒱^-∞_Γ_2(M)→
 
	extending the Alesker-Poincaré duality on the dense subspaces of smooth valuations.


It follows that if ϕ=[T,C] with C smooth, and X∈𝒫(M) s.t. [[N^*X]]∩(T^s)=∅, then one can naturally evaluate ϕ(X):=⟨ϕ, χ_X⟩. In particular, if X, Y∈𝒫(M) have disjoint conormal cycles, one can evaluate ∫_Mχ_X·χ_Y. In all reasonable settings, the result should equal χ(X∩ Y). This was shown to be the case when X,Y are transversal submanifolds with corners in <cit.>.

The Euler-Verdier involution σ:𝒱^-∞(M)→𝒱^-∞(M) was introduced by Alesker in <cit.>. It can be described through its action on the defining currents: σ [T, C]=[(-1)^na^*T, C].

We will need the following simple lemma, which we later use in the proof of Proposition <ref>.

	Take a generalized valuation ϕ∈𝒱^-∞(M^n) satisfying σϕ=(-1)^n+1ϕ and C(ϕ) smooth. 
	Assume that ϕ(F)=0 for all closed hypersurfaces F for which N^*(N^*F) is disjoint from (ϕ). 
	Assume moreover that M can be covered by open charts U_α≅^n, such that if K⊂ U_α is a smooth, strictly convex body, then N^*(N^*∂ K)∩(ϕ)=∅. Then ϕ=0.

As generalized valuations form a sheaf over M <cit.>, we may assume M=U_α=^n.

By <cit.>, valuations of the form μ(∙ - K), where K is a convex body with smooth support function and μ∈ℳ_c^∞(^n), span a dense subspace in 𝒱_c^∞(^n). It remains dense if we only take strictly convex K. Fix such μ and K.

By <cit.> and approximating ϕ by smooth valuations in the Hörmander topology with the same Euler-Verdier eigenvalue, we have 
(-1)^n+1ϕ(K)=σϕ(K)=(-1)^n(ϕ(K)-ϕ(∂ K)). By assumption, ϕ(∂ K)=0, and hence ϕ(K)=0. It follows that

    ⟨ϕ, μ(∙ - K)⟩ = ⟨ϕ, ∫_^nχ(∙∩ (K+x))dμ(x)⟩= ∫_^nϕ(K+x)dμ(x)=0

concluding the proof.






 §.§ Translation invariant valuations

We will need the following standard facts when we study valuations in the dual Heisenberg algebra in section <ref>.

The space of smooth translation invariant valuations ^∞(^n) can be defined as 𝒱^∞(^n)^tr. It is a Fréchet space. We will also consider generalized translation-invariant valuations, ^-∞(^n):=𝒱^-∞(^n)^tr.

The even/odd valuations are ^±, ±∞ which have eigenvalue ±1 under the antipodal map. The k-homogeneous valuations are those ϕ∈^±∞ that have eigenvalue λ^k under all rescalings of ^n by λ>0.

It is well-known that _0^-∞(^n)={χ}, and by Hadwiger's theorem <cit.>, _n^-∞(^n)={_n}. It follows from McMullen's work <cit.> that ^±∞(^n)=⊕_k=0^n_k^±∞(^n).

The line bundle over _k(^n) whose fiber over E is (E) is called the Klain bundle.
The Klain map : ^+,∞_k→Γ^∞(_k(V), (E)) is given by (ϕ)(E)=ϕ|_E. It is well-defined by Hadwiger's theorem, and injective by a theorem of Klain <cit.>. By <cit.>, it admits an injective extension : ^+,-∞_k→Γ^-∞(_k(V), (E)).  

Dual to the Klain section is the Crofton map, given by :ℳ^∞(_n-k(V))^tr→^+,∞_k(V), given by (μ)(K)=∫__n-k(V)χ(E∩ K)dμ(E). It follows from Alesker's irreducibility theorem <cit.> that  is surjective. It was shown in <cit.> that it admits a surjective extension :ℳ^-∞(_n-k(V))^tr→^+,-∞_k(V)

The relation to valuations on manifolds is as follows:
[<cit.>]
	𝒲_k^∞(M)/𝒲_k+1^∞(M) is naturally isomorphic to Γ^∞(M, _k^∞(TM)), the space of smooth sections of the bundle with fiber _k^∞(T_xM) over x∈ M. 
	
Consequently by the Alesker-Poincaré duality we have 
    𝒲_k^-∞(M)/𝒲_k+1^-∞(M)≃ (𝒲_n-k^∞(M)/𝒲_n-k+1^∞(M))^* ≃Γ^∞(M, _n-k^∞(TM))^*.





 §.§ Crofton formulas on manifolds
  

Consider a double fibration

    W[dl]_τ[dr]^π   
     Z        X
 where π, τ are proper, and τ×π: W→ Z× X is a closed embedding. We view Z as a space parameterizing a family of submanifolds ẑ:=πτ^-1z⊂ X, of dimension ẑ= W- Z.
Assume furthermore that the map Z→𝒟(ℙ_X), z↦ [[N^*ẑ]] is smooth. 

Given a distribution μ∈ℳ^-∞(Z), define the generalized valuation (μ)∈𝒱^-∞(X) by (μ)=π_*τ^*μ, which is the Radon transform with respect to the Euler characteristic of μ in the terminology of Alesker <cit.>. Explicitly, for ψ∈𝒱^∞_c(X),

    ⟨(μ),ψ⟩=∫_Zψ(ẑ) dμ(z)
  

	Let X be a manifold, and ϕ=(μ)∈𝒱^-∞(X) for μ∈ℳ^-∞(Z). Then ϕ∈𝒲_ X- W+ Z^-∞(X), and σϕ=(-1)^ X- W+ Zϕ, where σ is the Euler-Verdier involution.


We use the Alesker-Poincaré duality: for every smooth, compactly supported ψ∈𝒲_ W- Z+1^∞(X), it holds that ⟨ψ, (μ)⟩ = ∫_Z ψ(ẑ)dμ(z)=0. Now Proposition 7.3.2 in <cit.> implies the first statement.
	For the second statement, simply note that the defining current of ϕ is given by ∫_Z [[N^*ẑ]]dμ(z). 




§ DUAL HEISENBERG MANIFOLDS



 §.§ Geometric preliminaries

Recall that the Heisenberg algebra is ^2n+1 whose Lie bracket is given on the standard basis by [e_i, e_i+n]=e_2n+1 for 1≤ i≤ n, and [e_i, e_j]=0 for all i<j≠ i+n. The center of the algebra is thus Z:=(e_2n+1).

The dual Heisenberg algebra U^2n+1 is ^2n+1, equipped with a distinguished hyperplane H⊂ U, and ω∈∧^2 H^*⊗ U/H is a non-degenerate (twisted) form. Let us justify this terminology.


	The space U^* is naturally the Heisenberg Lie algebra.

Denote Z=(U/H)^*=H^⊥⊂ U^*. There is a Z-valued symplectic form on the quotient U^*/Z=H^*: we may consider ω as the isomorphism H≃  H^*⊗ (U/H), or equivalently H^*≃ H⊗ (U/H)^*, which is a non-degenerate bilinear form ω^*∈  (H⊗ H)⊗ (U/H)^*, which is clearly antisymmetric, that is ω^*∈∧ ^2 (U^*/Z)^* ⊗ Z. The Lie bracket [x, y]:=ω^*(x+Z, y+Z) then defines on U^* a Heisenberg Lie algebra structure with center Z.



	A manifold X equipped with a hyperplane distribution H and a smooth field of nowhere-degenerate forms ω∈Γ^∞(X, ∧ ^2 H_x^*⊗ (T_xX/H_x)) will be called a dual Heisenberg (DH) manifold. We call H the horizontal distribution and ω the DH form.


	Ovsienko <cit.> defines the closely related notion of a local Heisenberg structure. Similar ideas lie in the foundation of Stein's Heisenberg calculus, see e.g. Ponge <cit.>. 




	A contact manifold is naturally equipped with a DH structure. Let (X, H) be contact, and let α be a contact form defined locally. Then  α:T_xX/H_x is an isomorphism, and we may define the 2-form ω(u,v)=α^-1(dα(u, v)) for u, v∈ H_x. It is independent of the choice of α.



Let (X, H_X, ω_X) be a DH manifold. A DH-submanifold is a submanifold Y^2k+1⊂ X^2n+1 which is nowhere tangent to H_X. It inherits the structure of a DH manifold with hyperplane distribution H_Y=H_X∩ TY, and DH sform ω_Y=ω_X|_H_Y, where we naturally identify TX/H_X=TY/H_Y. 



	Let U^2n+1 be the dual Heisenberg Lie algebra, with horizontal hyperplane H⊂ U and DH form ω∈∧^2 H^*⊗ U/H. A Euclidean structure P on U is said to be compatible with the DH structure if one can find an orthonormal basis X_j, Y_j, Z of U such that X_j, Y_j form a symplectic basis of H w.r.t. Z: ω(X_i, Y_j)=0, ω(X_j, Y_j)=Z+H∈ U/H for all 1≤ i,j≤ n and i≠ j. A compatible form α∈ U^* is any form such that α=H. P is compatible with α if α=P(Z, ∙).



	Let M^2n+1 be a DH manifold with horizontal distribution H⊂ TM and DH forms ω∈∧ ^2 H^*⊗ TM/H. A Riemannian metric g on M is compatible if g_x is compatible for (T_xM, H_x, ω_x), for every x∈ M. 



	1. A similar notion of adapted metric was used by Chern and Hamilton in <cit.> for contact 3-manifolds and in later works by other authors.
	

	2. In fact, for the purposes of this work we only need the metric we choose to be compatible with the contact structure at isolated points of interest.



	For any DH manifold M, a globally defined compatible Riemannian metric exists. Moreover, if a compatible form α∈Ω^1 is given, one may choose the metric to be compatible with α.

Cover M by contractible open subsets U_i, and choose 1-forms α_i on U_i such that α_i|_U_i∩ U_j=±α_j|_U_i∩ U_j - this can be accomplished by fixing an arbitrary Riemannian metric and letting α_i(x)∈ T_x^*M have unit norm for all x∈ U_i. Set ω_i=α_i∘ω∈Γ(U_i, ∧^2 T^*M). Let us also fix a complementing line bundle L to H: L⊕ H=TM.

For each U_i we may choose a compatible complex structure J_i for ω_i on H|_U_i such that J_i|_U_i∩ U_j=± J_j|_U_i∩ U_j. We thus obtain the corresponding Euclidean forms h_i on H|_U_i which will satisfy h_i=h_j, and so can be patched to give a globally defined positive definite quadratic form h on H. Now for every U_i, choose Z_i∈Γ(U_i, L) satisfying α_i(Z_i)=1, and define the Riemannian metric g_i over U_i by g_i|_H=h, g_i(L, H)=0, g_i(Z_i, Z_i)=1. Then clearly g_i=g_j on U_i∩ U_j and thus (g_i) define a Riemmanian metric g on M. Over U_i we may locally choose an orthonormal symplectic basis X^i_j, Y^i_j for ω_i, and then ω(X^i_j, Y^i_j)=α_i^-1(1)=Z_i for all j. This shows g is a compatible metric.

Finally, if α is given, we just take α_i=α|_U_i in the construction above.



Let M^2n+1 be a DH manifold with horizontal distribution H and DH form ω. Set M_H:={(x, ξ): x∈ M, ξ⊥ H_x}⊂ℙ_M(=ℙ_+(T^*M)).


	
	A C^2 boundaryless submanifold F⊂ M is said to be normally transversal (to H) if N^*F intersects M_H transversally.
		
	More generally, a C^2 submanifold with boundary  F⊂ M is normally transversal if T_xF⊄H_x for all x∈∂ F, and both N^*(int F) and N^*∂ F intersect M_H transversally.


Note that if F is normally transversal then so is ∂ F.


	If a submanifold with boundary  F⊂ M intersects H normally transversally, then ([[N^*F]])∩ N^*M_H=∅.

For a boundarlyess submanifold this is immediate: N^*F is a boundaryless submanifold, and ([[N^*F]])=N^*(N^*F).

Now assume F has boundary.  Consider the sets N_∂ F^*F=∪_x∈∂ F N^*_xF and N_int^*F:=∪_x∈  F{ξ∈ℙ_+(T_x^*M): ξ∈ (T_xF)^⊥}. Both are manifolds with the same boundary B={(x,ξ): x∈∂ F, ξ∈ (T_xF)^⊥}, and N^*F=N_∂ F^*F∪ N_int^*F. Also, N_∂ F^*F⊂ N^*∂ F is a submanifold with the same boundary B.

By assumption, M_H intersects the interior points of N_∂ F^*F and N_int^*F transversally, and does not intersect B. It also holds for (x,ξ)∉ B that _x,ξ([[N^*F]]) coincides with N_x,ξ^*(N_∂ F^*F) if x∈∂ F, and with N_x,ξ^*( N_int^*F) if x∈intF. This concludes the proof.


Normal transversality is of course generic:

	Any hypersurface with boundary F⊂ M can be perturbed by an arbitrarily C^∞-small  amount to become normally transversal.

First, we may perturb F so that it is tangent to the contact distribution at isolated points.
Now near interior contact points, we only need to perturb F locally near those points to get normal transversality.
Next, we may perturb F near the boundary to have no contact points of F near the boundary, and isolated contact points of ∂ F.
We then perturb F near those contact points s.t. ∂ F has normal transversality at all its contact points.


Normal transversality is universal to embeddings, as the following lemma shows.

	Let (Y, H^Y)⊂ (X, H^X) be DH manifolds, and F⊂ Y a normally transversal submanifold with boundary. Then F⊂ X is also normally transversal.

We will distinguish the conormal bundles of F in the different ambient manifolds by writing e.g. N^*_YF⊂ℙ_Y. Recall that H^Y_x=T_xX∩ H^X_x. Thus for x∈∂ F, T_xF⊄H^X_x, since T_xF⊂ H^Y_x. 

Next take (x,ξ)∈ N_X^*F. For x∈int F we should check T_x,ξN^*_XF∩ T_x,ξX_H^X={0}. 
Consider the natural surjective map β:ℙ_X|_Y∖ N^*X→ℙ_Y. It holds that β(N^*_XF)=N^*_YF.
Denote X_H^X|_Y:=X_H^X∩π_X^-1(Y) where π_X:ℙ_X→ X. Since X⊂ Y is a DH submanifold, it holds that β:X_H^X|_Y→ Y_H^Y is a diffeomorphism. If v∈ T_x,ξN^*_XF∩ T_x,ξX_H^X|_Y then dβ(v)∈ T_x,ξY_H^Y∩ T_x,ξN^*_YF is non-zero, a contradiction to the assumption of normal transversality of F⊂ M.
The case of x∈∂ F is virtually identical.





 §.§ Constructing the valuations

Consider a point q=(p,ξ)∈ M_H. The vertical subspace is (dπ)=ξ^*⊗ T^*_pM/ξ=(H_p^⊥)^*⊗ H_p^*. Recall that ω_p∈∧^2 H_p^* ⊗ T_pM/H_p is non-degenerate, so that H_p^*≃ H_p⊗ H_p^⊥. Thus (dπ)≃ H_p.
Note also that dπ: T_qM_H→ T_pM is an isomorphism, so that we have a natural decomposition


    T_q ℙ_M≃ T_pM⊕ H_p

Denoting the contact hyperplane of ℙ_M by H_q, we get 

    H_q≃ H_p⊕ H_p.



We will define certain  η_k∈Γ^-∞_M_H(ℙ_M, Ω^2n+1⊗π^*o_M), 1≤ k≤ 2n (twisted by the pull-back of the orientation bundle of M) supported on M_H.

Take ψ̃∈Ω_c^2n+1(ℙ_M) a Legendrian form. We have 
    ψ̃|_q∈∧^2n+1T_q^*(ℙ_M)≃⊕_k=0^2n∧ ^2n+1-kT^*_pM⊗∧^k H_p^*

Moreover, since ψ̃ is Legendrian, its first factor in the k-th component belongs to the kernel of the restriction map ∧^2n+1-kT_p^*M→∧^2n+1-kH_p^*, which is naturally isomorphic to H_p^⊥⊗∧^2n-k H_p^*. Thus the k-th component of ψ̃, denoted  ψ̃^2n+1-k,k, lies in 
H_p^⊥⊗∧^2n-kH_p^*⊗∧^k H_p^*. 

Wedging the last two factors, we then get an element (ψ̃)_k∈∧^2nH_p^*⊗( T_pM/H_p)^* ≃∧ ^2n+1T^*_pM≃∧^2n+1T^*_q M_H. Now set

    ⟨η̃_k,ψ̃⊗ϵ_M ⟩ :=∫_M_H (ψ̃)_k ⊗ϵ_M

where ϵ_M is a section of π^* o_M, so that ψ̃_k ⊗ϵ_M∈(T_q M_H). Thus η̃_k is a linear functional on compactly supported, Legendrian, π^*o_M-twisted (2n+1)-forms, and it is supported on M_H. For k=0, η̃_0=[[M_H]] is in fact a well-defined (twisted) generalized 2n-form.

Finally for 1≤ k≤ 2n, define η_k∈Ω_-∞^2n+1(ℙ_M)⊗π^*o_M by setting for ψ∈Ω_c^2n(ℙ_M) 
    ⟨η_k, ψ⊗ϵ_M⟩:= ⟨η̃_k, Dψ⊗ϵ_M⟩
 where D denotes the Rumin differential. It is clear that η_k is a (twisted) Legendrian cycle, and π_*η_k=0.


	Define ϕ_k∈𝒱^-∞(M) for 0≤ k≤ 2n as follows: for 1≤ k≤ 2n by ϕ_k=[1/2η_k, 0]; for k=0, ϕ_0 is represented by (1/2η̃_0, 0).


	It is in fact possible to define natural curvature measures globalizing to ϕ_k: extend η̃_k arbitrarily as a functional on all forms, that is η̃_k∈Ω_-∞^2n(ℙ_M), and define the generalized curvature measure Φ_k represented by the pair of forms (1/2η̃_k, 0). The resulting curvature measure is independent of the extension, as η̃_k is only applied to subsets of conormal cycles.

Note that ϕ_0(X)=∫_N^*X1/2η̃_0=1/2∫_N^*X[[M_H]] is one-half the intersection index of M_H and N^*X, both oriented locally by a fixed local orientation on M. 
In the following proofs, we often assume for simplicity M is oriented. They are easily adjusted for the general case.

	ϕ_0 is the Euler characteristic.

Note that [[M_H]] is a closed current, and denoting π:ℙ_M→ M clearly π_*[[M_H]]=2. It follows that ϕ_0=χ by an obvious extension  to 𝒱^-∞ of <cit.>.



Recall that a:ℙ_M→ℙ_M is the fiberwise antipodal map, and σ:𝒱^-∞(M)→𝒱^-∞(M) denotes the Euler-Verdier involution.

	For all 0≤ k≤ 2n it holds that σϕ_k=ϕ_k.

For ψ∈Ω^2n+1(ℙ_M) it holds that ∫_M_Hψ=∫_M_Ha^*ψ, since a:M_H→ M_H is clearly orientation preserving. Note also that a:ℙ_M→ℙ_M is orientation reversing as the fibers of ℙ_M→ M are even-dimensional spheres. Hence for a test form ψ∈Ω_c^2n(ℙ_M),

    ⟨ a^*η_k, ψ⟩=- ⟨η_k, a^*ψ⟩=-∫_M_H[Da^*ψ]_k=-∫_M_Ha^*[Dψ]_k=-∫_M_H[Dψ]_k=-⟨η_k, ψ⟩

That is, a^*η_k=-η_k for all k, and hence 
σϕ_k=(-1)^2n+1(-ϕ_k)=ϕ_k as asserted.



	It holds that ϕ_k∈𝒲_k^-∞(M).

Take a closed Legendrian form ψ∈Ω^2n+1(ℙ_M) defining a smooth valuation Ψ∈𝒲_2n+2-k^∞(M). We ought to show that ϕ_k·Ψ=0, equivalently ⟨η̃_k, ψ⟩=0.
But Ψ∈𝒲_2n+2-k^∞(M) implies that ψ has a horizontal degree at least 2n+2-k, and the claim follows from the definition of η̃_k.



	The wavefront set of ϕ_k for 1≤ k≤ 2n is N^*M_H.

This is immediate from definition: the wavefront set of [[M_H]] is N^*M_H, and restriction to M_H is the only source of singularities of η_k.
It follows by Lemma <ref> that we may evaluate ϕ_k on any normally transversal submanifold with boundary.

Let F⊂ M be a smooth hypersurface in a DH-manifold M^2n+1, assume that F is normally transversal to the horizontal distribution, and tangent to it at p∈ M.

Working in a small open ball U near p with no other contact points, let us fix a 1-form α∈Ω^1(U) defining the horizontal distribution, which also trivializes the DH form: ω∈Γ(U,∧^2 H^*). Let g be a Riemannian metric on U which is compatible with α and ω on T_pM (but not necessarily elsewhere), with Levi-Civita connection ∇. Let R:U→ SM be the vector field given by R(x)⊥ H_x,  g(R, R)=1, α(R)>0. Let (X_j)_j=0^2n be an orthonormal frame in U with X_0|_F=ν:F∩ U→ SM the unit normal oriented by α(ν)>0, and (X_j(p))_j=1^2n a symplectic basis of H_p. Let θ_j∈Ω^1(U) be the dual coframe to X_j. Let S=(s_ij)_i,j=1^2n, s_ij=θ_i(∇_X_jν) be the second fundamental form given by ∇_X_jν = ∑_i=0^2ns_ij X_i. 

	Let h_ij, 1≤ i≤ 2n, 0≤ j≤ 2n be given by h_ij= θ_i(∇_X_jR).    
	The matrix h=(h_ij)_i,j=1^2n is the second fundamental form of the contact structure.


This is a slightly different definition than the one in <cit.>, where the second fundamental form is symmetrized.

Define the matrix A_p=(h_ij-s_ij)_i,j=1^2n.



	
	Let F be a normally transversal closed hypersurface. It then holds that
	
	
    ϕ_k(F)= 2n  k∑_p: T_pF=H_p | A_p|^-1D(A_p[2n-k], J[k])

	where D is the mixed discriminant.
	


	We thus see that geometrically, ϕ_k is reminiscent of the k-th elementary symmetric polynomial in the principal radii of a hypersurface in a Riemannian manifold. In particular, ϕ_2n^-1 plays the role of the absolute value of the gaussian curvature. In fact, for M=U - the dual Heisenberg algebra, ϕ_2n(F) is precisely the inverse absolute value of the gaussian curvature, summed over all contact points.
	

Write ω_i for the Ehresmann connection on SU, namely ω_i=π_V^*θ_i, where π_V:T_p,ξSU→ T_p M is the projection to the vertical tangent space.
It then holds that R^*ω_i=∑_j=0^2n h_ijθ_j, and  ν^*ω_i=∑_j=1^2ns_ijθ_j. Recall that (s_ij) is symmetric. For notational simplicity, we write θ_j also for π^* θ_j∈Ω^1(SU).

First we describe the normal cycle NF={(x, ±ν(x)): x∈ F}⊂ SM near (p,ξ)=X_0(p)∈ M_H, explicitly as a generalized form ω_NF:=[[NF]]∈Ω_-∞(SM). 
Let W⊂π^-1U be a small neighborhood of (p, ξ).
Write σ_SM:=∧_i=0^2nθ_i ∧∧_i=1^2nω_i, and θ_F=∧_i=1^2nθ_i∈Ω^2n(F∩ U).

 Claim. ω_NF^W:=ω_NF|_W∈Ω_-∞(W) is given by

    ω_NF= θ_0∧∧_i=1^2n(ω_i-∑_j=1^2ns_ijθ_j)δ_NF^W

where


    ⟨δ^W_NF, μ(x,ζ)σ_SM⟩=∫_F∩π(W)μ(x, ν(x)) θ_F


Proof of Claim.
Take ψ∈Ω^2n_c(W). We should check that 


    ∫_NFψ= ∫_F ν^*(ψ∧θ_0∧∧_i=1^2n(ω_i-∑_j=1^2ns_ijθ_j)/σ_SM) θ_F

which reduces to the pointwise verification

    ν^*ψ/θ_F=ν^*(θ_0∧ψ∧∧_i=1^2n(ω_i-∑_j=1^2ns_ijθ_j)/σ_SM)

We will check this equality for a basis of the 2n forms, which is given by 2n-wedges of (θ_i)_i=0^2n, (ω_j)_j=1^2n. 
If ψ contains a θ_0 factor, clearly both sides vanish.

Assume ψ = θ_1∧…∧θ_k∧ω_k+1∧…∧ω_2n.
The left hand side is easily seen to be equal (s_ij)_i,j=k+1^2n. The right hand side is 

    ν^*((-1)^2n-k(s_ij)_i,j=k+1^2nθ_0∧…∧θ_k∧ω_k+1∧…∧ω_2n∧ω_1∧…∧ω_k
    	∧θ_k+1∧…∧θ_2n/σ_SM)

and after reordering we get (-1)^2n-k+k(2n-k)(s_ij)_i,j=k+1^2n=(s_ij)_i,j=k+1^2n, concluding the proof of the claim.


We wish to apply η̃_k to ω_NF^W. Since NF intersects M_H transversally at the isolated point p, we need only look at the value of ω_NF^W at (p,± X_0(p)).

Recall the decomposition T_p,ξℙ_M=T_p,ξM_H⊕ T_ξ(ℙ_+(T^*_pM)).
We denote by W_p,ξ=T_p,ξM_H=T_pM the contact-horizontal subspace, and by V_p,ξ=T_ξ(ℙ_+(T^*_pM))=(H_p⊗ H_p^⊥)^* the vertical subspace.
Using the Riemannian structure, we get a decomposition T_p,ξ(SM)=W_p,ξ⊕ V_p,ξ.
The DH form ω∈∧^2 H_p^*⊗ T_pM/H_p can be recast as the isomorphism Ω:H_p⊗ H_p^⊥→ H_p^*, that is Ω:V_p,ξ^*→ H_p^*.

We first need to identify the (2n+1-k, k) component (with respect to decomposition (<ref>)) of ω_NF^W over (p, ξ)∈ M_H. 
The forms θ_i vanish on the vertical subspace.
The contact-horizontal component of ω_i is given by R^*ω_i=∑_j=0^2n h_ijθ_j. We will write the resulting decomposition as ω_i=R^*ω_i⊕ω_i^V
where R^*ω_i is the contact-horizontal component, while ω_i^V, the restriction of ω_i to  V_p,ξ, is the vertical component. 

Define (γ_ij)_i,j=1^2n by Ω(ω_i^V)=∑_j=1^2nγ_ijθ_j∈ H_p^*. Note that under the Euclidean identification, V_p,ξ=H_p and ω_i^V=θ_i for i=1,…,2n.  Compute γ_ij=Ω(θ_i)(X_j)=ω(X_i, X_j). By assumption, X_j is a symplectic basis of H_p, so that γ_ij is the standard 2n× 2n matrix J representing √(-1). 


Write a_ij=h_ij-s_ij, 1≤ i,j≤ 2n. Then 


    ω_NF^W= δ_NF^W·θ_0∧∧_i=1^2n (ω_i^V+∑_j=1^2na_ijθ_j)
 so that

    (ω_NF^W)^2n+1-k,k=δ_NF^W·θ_0∧∑_|I|=kϵ_I ∧_i∉ I(∑_j=1^2na_ijθ_j)∧∧_i∈ Iω_i^V

where ϵ_I=(-1)^i_2-i_1+… + i_2j-i_2j-1-j is the sign of the permutation σ_I=1… 2n  I^c, I, where I={i_1<… <i_k}, I^c={1,…,2n}∖ I is ordered increasingly, and j=⌊k/2⌋.

Applying Ω to the last k factors and subsequently wedging all the factors to get a top form on T_pM, we get


    ⟨η̃_k,  ω_NF^W⟩   =∫_M_Hθ_0∧∑_|I|=kϵ_I ∧_i∉ I(∑_j=1^2na_ijθ_j)∧∧_i∈ I(∑_j=1^2nγ_ijθ_j)·δ_NF ^W 
       
    = ∑_|I|=k∑_σ∈ S_2nσ∏_i∉ Ia_iσ(i)∏_i∈ Iγ_iσ(i)∫_M_Hθ_0∧…∧θ_2n·δ_NF^W


Recall A_p=(a_ij)=h-S where S=(s_ij), h=(h_ij), 1≤ i,j≤ 2n, and let B^α be the α-row of a matrix B. For an ordered subset I⊂{1,…,2n}, let (A_p^I^c, J^I)  denote the matrix with the corresponding columns. Then

    ⟨η̃_k,  ω_NF^W⟩= ∑_|I|=k (A_p^I^c, J^I)∫_M_Hθ_0∧…∧θ_2n·δ_NF^W


Define 
    B(α)={[    A_p, α≤ 2n-k;      J,  α>2n-k ].
 The mixed discriminant is given by


    D(A_p[2n-k], J[k])=1/(2n)!∑_τ∈ S_2n( B(τ(i))_i,j)=1/(2n!)∑_|I|=k k!(2n-k)! (A_p^I^c, J^I)
 
that is

    ⟨η̃_k, ω_NF^W⟩=2n k D(A_p[2n-k], J[k]) ∫_M_Hθ_0∧…∧θ_2n·δ_NF^W


Recall that we should fix a section of the orientation bundle of M over SM to get numerical values for the integral. 
Let us verify that for a choice of ϵ_θ∈π^*o_M corresponding to θ_0∧…∧θ_2n we get the identity 

    ∫_M_Hθ_0∧…∧θ_2n·δ_NF^W⊗ϵ_θ=| A_p|^-1.

One can compute it directly, but we can do something simpler: Observe that ⟨η̃_0, ω_NF^W⟩ is just the intersection index I_p,ξ of NF and M_H at (p,ξ). Note that the order of intersection is not important as NF is even, while the orientations of M_H and NF are determined by ϵ_θ.


By what we have proved, we see that 
    I_p,ξ(NF,M_H)=⟨η̃_0,  ω_NF^W⟩ = A_p ∫_M_Hθ_0∧…∧θ_2nδ_NF^W.

It thus remains to verify that I_p,ξ(NF, M_H)= A_p.

The positive orientation on T_p, ξSM is given by the dual basis θ_0,…, θ_2n, ω_1, …,ω_2n. To see that, consider a homotopy of the Riemannian metric between our metric and a flat one, and some corresponding homotopy of the orthonormal frame X_j. The dual basis above remains a basis throughout the homotopy, and clearly defines the positive orientation in the flat case.  

Considering ν:F→ SM, R:F→ SM as maps, we get a positive basis of T_p,ξM_H given by D_pR(X_0),…, D_pR(X_2n), and a positive basis of T_p,ξNF given by D_pν(X_1),…, D_pν(X_2n).


Form the (4n+1)× (4n+1) matrix 


    B=([ θ_i(D_pR(X_j))_i=0… 2n^j=0… 2n θ_i(D_pν(X_j))_i=0… 2n^j=1… 2n; ω_i(D_pR(X_j))_i=1… 2n^j=0… 2n ω_i(D_pν(X_j))_i=1… 2n^j=1… 2n ]) =  ([ δ_ij δ_ij; h_ij s_ij ])

By definition, I_p,ξ(NF,M_H)= B.
Now for 1≤ i≤ 2n substract column 2n+1+i from column i+1. Finally, interchange each culumn of index 2… 2n+1 with the respective column from the last 2n columns, resulting in 4n^2 swaps. The resulting matrix is block-triangular, and has determinant B= (h_ij-s_ij)_i=1… 2n^j=1… 2n= A_p, verifying our assertion.

Accounting also for the point ξ=H^⊥ with the opposite orientation, we conclude that


    ⟨η̃_k, ω_NF⟩ =2 2n  k∑_p: T_pF=H_p | A_p|^-1D(A_p[2n-k], J[k])

and the statement follows.

 For a normally transversal contact point p of F, the local contact areas are 
    ϕ_k(F, p):= 2n  k | A_p|^-1D(A_p[2n-k], J[k]).




	When k=2n, equation (<ref>) remains valid also for a normally transversal hypersurface with boundary F.

By Lemma <ref>, ϕ_2n(F) is well-defined. Using the notation of the proof of Lemma <ref>, we may write 

    ϕ_2n(F)=1/2⟨η̃_2n, [[N^*_intF]] ⟩+1/2⟨η̃_2n, [[N^*_∂ FF]]⟩.

 The first summand is computed in Proposition <ref>. The second summand trivially vanishes.



For a normally transversal hypersurface with boundary F, ϕ_2n(F)≥ 0,  with equality if and only if it is nowhere tangent to H.

Immediate from Corollary <ref> and eq. (<ref>) .
It will be convenient to extend ϕ_2n to general hypersurfaces by setting ϕ_2n(F)=∞ when F is not normally transversal.
 
Next we establish the universality of ϕ_k with respect to embeddings.

Let (X^2n+1, H^X, ω) be a DH-manifold, and Y^2m+1⊂ X^2n+1 a DH-submanifold. Let i:Y→ X denote the embedding. We will write η_2k^X, ϕ_2k^X for the generalized forms and valuations defined over X, and similiarly for Y. 


	It holds that i^*ϕ_k^X=ϕ_k^Y.

The following constructions appeared in <cit.>, and we refer therein for more details. Consider the natural submersion β:ℙ_X|_Y∖ N^*Y↠ℙ_Y and the inclusion α:ℙ_X|_Y↪ℙ_X. Let π̃:W→ℙ_X|_Y be the oriented blow-up of ℙ_X|_Y along the conormal bundle N^*Y⊂ℙ_X, and α̃: W→ℙ_X the corresponding lift. Let β̃:W→ℙ_Y be induced by the restriction map X× _Y T^*Y→ T^*X. For a valuation Ψ∈𝒱^∞(Y) defined by the closed Legendrian form ψ∈Ω^2m+1(ℙ_Y), i^*Ψ is given by the current α̃_*β̃^*ψ.  
Now ⟨ϕ_k^Y, Ψ⟩= ⟨η̃_k^Y, ψ⟩ and similarly for X. We should thus verify that 

    ⟨η̃_k^Y, ψ⟩=⟨η̃_k^X,α̃_*β̃^*ψ⟩

Note that η̃_k^X is supported on X_H^X⊂ℙ_X, which by assumption is disjoint from N^*Y. Hence the right hand side can be replaced by ⟨η̃_k^X,α_* β^*ψ⟩.

Next take p∈ Y, ξ=(H^Y_p)^⊥∈ℙ_Y, and (p,ξ')∈ℙ_X|_Y s.t. β(p,ξ')=(p,ξ). We may decompose T_p,ξ'ℙ_X|_Y=T_pY⊕ (H_p^X)^*⊗ (H_p^X)^⊥*. Consider dβ: T_p,ξ'ℙ_X|_Y→ T_p,ξℙ_Y, so that

    dβ^*: T^*_pY⊕ H_p^Y⊗ (T_pY/H_p^Y)^*→ T^*_pY⊕  H_p^X⊗ (T_pX/H_p^X)^*

acts as the identity on the first summand. On the second summand, recalling we have the canonical identificaion T_pY/H_p^Y≃ T_pX/H_p^X induced by the inclusion T_pY⊂ T_pX, dβ simply acts as the inclusion H_p^Y↪ H_p^X. 
Let us denote by Ω_k^Y: ∧^k H^Y_p⊗ (H^Y_p)^⊥→∧ ^k (H^Y_p)^* the isomorphism induced by the DH form ω, and similarly for X. It follows that the following diagram commutes:



	H_p^Y⊗(H_p^Y)^⊥[r]^–dβ^*[d]^Ω_Y      H_p^X⊗(H_p^X)^⊥[d]^Ω_X

	(H_p^Y)^*    ( H_p^X)^*[l]^res 

We are left with verifying the identity


    ∫_Y_Hψ_0⊗ϵ_Y= ∫_X_Hα_*β^*ψ_0⊗ϵ_X
 for an arbitrary Legendrian form ψ_0∈Ω^2m+1(ℙ_Y).
But this is now equivalent to the statement i^*ϕ_0^X=ϕ_0^Y, which holds as both sides are just the Euler characteristic.




Weyl's principle for DH manifolds, which we just established, is readily applicable in conjunction with the following technical lemma.


	Consider a compact submanifold with boundary F^2k⊂ M^2n+1 of a DH manifold (M, H, ω). 
	
		
  * If 2k≤ n, F lies in fact inside a DH submanifold N^2k+1⊂ M. 
		
  * For arbitrary k<n, one may find a pair of DH-manifolds N^2k+1⊂ X^4n+1 such that we get a commuting diagram of DH manifolds
	
		
		    N^2k+1@^(->[rd]     
  F^2k @^(->[ru] @^(->[r]    M^2n+1@^(->[r]     X^4n+1    
			 
		
		where all inclusions are DH-embeddings. 
		
	


Assume first 2k≤ n. Choose a Riemannian metric g on M, and let L:F→ℙ(TM)|_F be given by L(x)=H_x^⊥, the orthogonal complement with respect to g. Note that L(F) does not intersect ℙ(H)|_F⊂ℙ(TM)|_F. By the transversality theorem, and since 
    L(F)+ℙ(TF)<ℙ(TM)|_F 2k+4k-1<2n+2k 2k≤ n,
 we may perturb g so that L(F) avoids ℙ(TF)⊂ℙ(TM). We then take N to be the image under the exponential map of a small neighborhood of the zero section in TF⊕ L(F)⊂ TM|_F. It is clearly a DH-submanifold containing F.

Now in the general case, consider M_H⊂ℙ(T^*M), which is the quotient of M_H⊂ℙ_M under the two-covering map ℙ_M→ℙ(T^*M). Define a DH structure on a neighborhood of M_H as follows: the horizontal structure H will be the canonical contact structure of ℙ(T^*M). For (x,ξ)∈M_H, by eqns. (<ref>) and (<ref>), H_x,ξ is canonically identified with H_x⊕ H_x⊂ T_xM⊕ H_x. Noting that T_x,ξℙ(T^*M)/H_x,ξ≃ T_xM/H_x, define ω_X∈∧^2H_x,ξ^*⊗ T_x,ξℙ(T^*M)/H_x,ξ by 
    ω_X((u_1,v_1), (u_2, v_2))=ω(u_1, v_1)+ω(u_2, v_2)  ∀ (u_1, v_1), (u_2, v_2)∈H_x,ξ.
 Extend ω_X arbitrarily to a global section of ∧ ^2H^*⊗ Tℙ(T^*M)/H. Take X to be a small neighborhood of M_H in which ω_X is non-degenerate. Then (X, H, ω_X) is a DH manifold, and the DH-submanifold M_H⊂ X is clearly isomorphic to M. It remains to note that X=2·2n+1 and 2k< 2n, and so we may choose the desired N^2k+1⊂ X by the first case.




	Let F^2k⊂ M be a submanifold with boundary, normally transversal to the horizontal distribution H. Then ϕ_2k(F)≥0, with equality if and only if it is nowhere tangent to H.

This follows at once from Lemmas <ref> and <ref>, Corollary <ref> and Theorem <ref>. 


Finally, we consider the relations between the valuations we constructed.


	(i) On any DH manifold,  ϕ_2b∈𝒲^-∞_2b∖𝒲^-∞_2b+1 as 0≤ b≤ n. In particular, ϕ_2b, 0≤ b≤ n are linearly independent valuations.
	
(ii) On a generic DH manifold (in a sense to be made precise in the proof), (ϕ_b)_b=0^2n are all linearly independent.



By Proposition <ref>, ϕ_k∈𝒲_k^-∞(M). For a normally transversal hypersurface F write 
    ϕ_k(F,p)   =| A_p|^-1D(A_p[2n-k], J[k])
       =(-1)^k| A_p|^-1∑_j=0^2n-k2n-k  j D(S_p[2n-k-j], h_p[j], J[k])
 

For even k=2b, the term of highest order in S_p (corresponding to j=0) is non-zero for generic S_p. In particular, ϕ_2n(F)≠ 0 for generic hypersurfaces F. 
Now choose any DH submanifold N⊂ M of dimension 2b+1. Using Theorem <ref>, Lemma <ref> and the last observation, we may find a 2b-dimensional submanifold F^2b⊂ N s.t. ϕ^M_2b(F)=ϕ^N_2b(F)≠ 0. Thus ϕ_2b∉𝒲_2b+1^-∞(M), proving (i).


For (ii), it now suffices to check that ϕ_2b-1, ϕ_2b are linearly independent in 𝒲^-∞_2b(M)/𝒲^-∞_2b+1(M), for a generic M.  In turn this is implied by the following statement: as a function on normally transversal closed submanifolds of dimension 2b lying inside a fixed DH submanifold N^2b+1⊂ M , ϕ_2b-1, ϕ_2b are linearly independent. By Theorem <ref>, we may assume b=n. Now note that for ϕ_2n-1, the summand of degree 1 in S_p vanishes, as S_p is symmetric while J is antisymmetric. The statement now follows by examining the summand whose numerator contains no S_p for both valuations, which has different coefficients (depending on p for a general horizontal distribution).





 §.§ Extending ϕ_2k to arbitrary 2k-submanifolds

Define the contact area _2k(F)∈[0,∞] for any C^2, 2k-dimensional submanifold with boundary F⊂ M by 


    _2k(F) : = lim inf_F_ϵ→ Fϕ_2k(F_ϵ)

where F_ϵ is a C^2, normally transversal submanifold with boundary that C^2-converges to F. 
Note that Proposition <ref> and Theorem <ref>, and the proof of Lemma <ref>, imply that ϕ_2k(F) is C^2-continuous on normally transversal submanifolds F. Hence on such submanifolds, _2k(F)=ϕ_2k(F).

We now show that the vanishing of _2k(F^2k) is a necessary condition for the existence of an arbitrarily small perturbation with no contact points. For closed F this supplements the topological necessary condition χ(F)=0.


	For a 2k-dimensional submanifold with boundary F⊂ M, _2k(F^2k)=0 if and only if there is an arbitrarily small C^2-perturbation of F which is nowhere tangent to the horizontal structure.

The if direction follows from Corollary <ref>. For the other direction, let us assume _2k(F)=0. Again by Corollary <ref>, F cannot be normally transversal to the horizontal distribution unless it is nowhere tangent to it. Assuming the contrary to the assertion, we conclude F is not normally transversal. Let F_ϵ→ F, ϵ→ 0 be a normally transversal family of smooth perturbations of F s.t. ϕ_2k(F_ϵ)→ 0. If a normally transversal contact point exists for F (which is then necessarily isolated), it persists to F_ϵ for small ϵ, and examining eq. (<ref>), one can find c>0 such that ϕ_2k(F_ϵ)≥ c for all small ϵ, a contradiction. Thus F is not normally transversal to the horizontal distribution at any of its contact points. It follows that a sequence x_j of contact points of F_ϵ_j must approach a necessarily degenerate tangency point, and hence by eq. (<ref>), ϕ_2k(F_ϵ_j)→∞. This again is a contradiction, implying that for small ϵ, F_ϵ has no horizontal tangent spaces, as claimed.


§ THE DUAL HEISENBERG LIE ALGEBRA





 §.§ Linear algebra





Recall that U^2n+1 denotes the dual Heisenberg algebra, with H⊂ U a fixed linear hyperplane, and ω∈∧^2 H^*⊗ U/H is a non-degenerate (twisted) form. It is the simplest DH manifold.

Let the group of automorphisms of the Heisenberg algebra be denoted by _H(U) or _H(2n+1). 
There is an _H(U)-equivariant isomorphism (H)≃(U/H)^n.

Define the subgroups _H^+(U)={ g∈_H(U): g|_U/H>0 }, ^1_H(U)={ g∈_H(U): g|_U/H=} and Scal_H(U)={g∈_H(U): g|_H∈^*}⊂_H^+(U). Note that for g∈_H^1(U) clearly g =1.
For an _H(U)-module X and x∈ X, we will write (x), resp. ^+(x), ^1(x) for its stabilizer in the corresponding subgroup. 
Let us record the following trivial fact. 

	_H^+(U) is generated by  ^1_H(U) and Scal_H(U)


	Let δ_λ∈Scal_H(U) act by λ∈ on H and by λ^2 on a fixed direction L complementing H. Then _H^1(U) and (δ_λ)_λ≠ 0 generate _H^+(U)



Take any g∈Scal_H(U), set λ:=g|_H. Then δ_λ^-1∘ g∈_H^1. The claim now follows from Lemma <ref>.




	For any subspace E⊂ H, set E_0=E∩ E^ω. Then one can find:
	
		
  * S∈^1(E) s.t. S(E_0)=E_0 and S|_E_0=2.
		
		
  * T∈^+(E) s.t. T(E_0)=E_0, T|_E_0=1 and T< 1.
	
	


i)
Decompose  E∩ H=E_0⊕ F, where F⊂ H is a non-degenerate subspace. Fix a vector z∈ E∖ H. There is then an induced symplectic form ω_z on H given by ω_z(u,v)(z+H)=ω(u,v). Define S∈(E)  by setting S|_E_0=2, S|_F=1, and note that S leaves ω_z|_E invariant. By Witt's extension theorem, we can find an extension  S∈(H, ω_z), and finally setting S(z)=z yields S∈^1(E) as required.



ii)  Using S from i), define T∈_H^+(U) by T|_H=1/2 S|_H, Tz=1/4 z.





The orbits Y^k_ϵ, r of _H^+(U) in _k(U) are classified by the pairs (ϵ, r) as follows: 
    E∈ Y^k_ϵ, rϵ=k- H∩ E∈{0,1},2r=κ- (ω|_E∩ H)
 where ϵ∈{0,1}, and  r∈{ 0, …, ⌊κ/2⌋}, κ =min (k-ϵ, 2n-(k-ϵ)). . The unique open orbit has ϵ=1 and r=⌊κ/2⌋, the unique closed orbit has ϵ=0 and r=0.



 §.§ Translation invariant valuations on the dual Heisenberg algebra


We start by classifying the _H^+(U)-invariant Klain sections. 


 Fix 1≤ k≤ 2n.
	
	i) For even k, there is at most a one-dimensional space of _H^+(U)-invariant generalized Klain sections over _k(U).
	
ii) For odd k, there are no _H^+(U)-invariant generalized Klain sections over _k(U).


We will make repeated use of Lemma <ref> without explicit mention, wherein also the bundle F_Y^α of principal symbols transversal to Y is defined.

Take E∈ Y^k_ϵ,r, and denote  E_0=(E∩ H)∩ (E∩ H)^ω, E_0=κ-2r.

Step 1. ϵ=1. We consider first the open orbit: r=r_max. If k is odd,  E∩ H is non-degenerate, and ω gives an isomorphism (E∩ H)=(U/H)^k-1/2. Thus

    (E)   =(E/E∩ H)⊗(E∩ H) = (U/H)⊗(U/H)^k-1/2
       =(U/H)^k+1/2=(U)^k+1/2(n+1).

It follows that for odd k there are no invariant sections on the open orbit.

For even k, let E_0⊂ E∩ H be the kernel of ω|_E∩ H, which is a line. Then


    (E)   =(E/E∩ H)⊗(E∩ H) 
       =(U/H)⊗(E_0)⊗((E∩ H)/E_0)
        = (U/H)^k/2⊗(E_0)=(U)^k/2(n+1)⊗(E_0)


By Lemma <ref>, we may find S∈^1(E) acting trivially on the first factor and rescaling the second factor. It follows that there are no invariant sections on the open orbit Y^k_1, r_max.

Now let us consider the orbit Y=Y^k_1,r with r<r_max. We have N_E Y=N_E∩ H X^k-1_r(H)=∧^2E_0^*. Consider the bundle over Y with fiber 
    F_Y^α|_E=(E)⊗^*(N_E Y)⊗^α (N_E Y)
    =(E)⊗(∧^2 E_0)⊗^α(∧^* E_0)

Since E/E∩ H is (E)-isomorphic to U/H, 


    (E)=(E∩ H)⊗(U/H)=((E∩ H )/ E_0)⊗(E_0)⊗(U/H)

Now (E∩ H )/ E_0=2r, and ω readily yields a non-degenerate form 
    ω̃∈∧^2((E∩ H )/ E_0)^*⊗ U/H
 so that there is a (E)-isomorphism ((E∩ H)/E_0)=(U/H)^r. Thus


    F^α_Y|_E=(U/H)^r+1⊗(E_0) ⊗(∧^2 E_0)⊗^α(∧ ^2 E_0^*),

Again by Lemma <ref>, there are no ^+(E)-invariants in F_Y^α|_E when r<r_max (and so E_0≠{0}). 

We conclude that for no k are there invariant generalized sections whose support intersects Y^k_1,r, for any r.
We assume from now on that ϵ=0, so E⊂ H.

Step 2. Consider Y=Y^k_0,r with r=r_max. Then N_E Y=E^*⊗ U/H, and 
    F^0_Y|_E=(E)⊗^*(N_EY) = 
    (E)^2⊗^*(U/H)^k

Taking g∈Scal_H(U) with g|_H=λ, we see that it acts on F^0_E⊗^α(N_EY)=(E)^2⊗^*(U/H)^k⊗ (U/H)^α⊗^α E^* by λ^-2k+2k+2α-α=λ^α.  Thus α=0 is the only possible transversal order of an invariant section. We now consider separately the different parities of k.

If k is odd, set E_0=E∩ E^ω, E_0=1. Then
 

    F_E^0= (E_0)^2⊗(E/E_0)^2⊗^*(U)^k/n+1 =(E_0)^2⊗^*(U)^k+1/2(n+1)

since (E/E_0)^2≃(U/H)^k-1= (U)^k-1/n+1. Thus by Lemma <ref>, the action of ^+(E) on F^0_E is clearly non-trivial, and so there are no invariant sections whose support intersects Y.

If k is even, the restriction of ω to E gives an isomorphism (E)=(U/H)^k/2, so that (E) acts trivially on F_E^0. We know by now that all invariant sections are supported inside Y. Thus we conclude that the space of restrictions of the space of invariant sections to _k(U)∖ (Y∖ Y) is at most one-dimensional.

Step 3. It remains to show there are no invariant sections supported on the closure of either of the orbits Y_0, r^k(U) with r<r_max, for any k. In particular, we have 1<k<2n and κ-2r= E_0≥ 2, with κ=min(k, 2n-k).

One has the chain of inclusions T_EY^k_0,r(U)=T_EX^k_r(H)⊂ T_E_k(H)⊂ T_E _k(U).  Hence N_E Y^k_0,r(U) fits into the exact sequence 

    0→ T_E_k(H)/T_EX^k_r(H)→  N_E Y^k_0,r(U) → T_E _k(U)/T_E_k(H)→ 0

which is (E)-isomorphic to

    0→∧ ^2 E_0^*→  N_E Y^k_0,r(U) → E^*⊗ U/H→ 0

Write Y=Y^k_0,r. For α=0, 

    F^0_Y|_E   =(E)⊗^*(N_EY^k_0,r(U))
       =(E_0)^2⊗(E/E_0)^2 ⊗^*(U/H)^k⊗^*(∧ ^2 E_0^*)
       =(E_0)^2⊗^*(U/H)^k-r⊗^*(∧ ^2 E_0^*) .

Thus (F^0_Y|_E)^^+(E)={0} since one can find g∈^1(E) with g|_E_0=λ≠1.


When α>0, a ^+(E)-invariant element of F^α_Y|_E would imply the existence of an invariant element in 

    F^0_Y|_E⊗ (∧ ^2 E_0^*)^a⊗ (E^*)^b⊗ (U/H)^b

which in turn implies the existence of an invariant element in


    (E_0)^2⊗(∧ ^2 E_0)⊗ (∧ ^2 E_0^*)^a⊗ (E_0^*)^b'⊗  ((E/E_0)^*)^b”⊗(U)^λ

for some non-negative integers a, b, b', b” and λ∈.

By the proof of Lemma <ref>, we can find S∈^1(E) such that S|_E_0=2 and S:E/E_0→ E/E_0 is the identity. Thus there are no invariants in this space.






For a vector bundle E over a manifold B, we let Γ_m(B, E):=Γ_c(B, E^*⊗ |ω_B|)^* denote the space of generalized sections that are given locally by a regular Borel measure.  


	There are _H^+(U)-invariant generalized Klain sections κ_k realizing the upper bounds obtained in Proposition <ref>. Moreover,  κ_k∈Γ_m(_k(U), (E)), they are supported on Y^k_H:={E⊂ H} and are _H(U)-invariant.

Take f(E)=|ω^∧k/2|^⊗ 2∈(E)^2⊗^*(U/H)^k, which is a continuous section over _k(H). We may rewrite f as an absolutely continuous measure on _k(H) with values in the bundle ^2(E)⊗^*(U/H)^k⊗^*(T_E_k(H))=^*(U)^k⊗(E)^2n+2. Writing i:_k(H)→_k(U) for the natural embedding, we get 
    i_*f    ∈ℳ__k(H)(_k(U), ^*(U)^k⊗(E)^2n+2)
       ≃Γ_m(_k(U), ^*(U)^k⊗(E)^2n+2⊗(T_E_k(U)))

The latter bundle is just the Klain bundle:

    ^*(U)^k⊗(E)^2n+2⊗(E^*⊗ U/E)=(E)

and it remains to note that we only used _H(U)-equivariant identifications.




Let us fix an involution R∈_H(U) acting by -1 on U/H. Since R^-1_H^+(U)R=_H^+(U), it follows that 
R acts on the space of _H^+(U)-invariants in any _H(U)-module M. We will call the ±1 eigenspaces of R in M^_H^+(U) R-even, resp. R-odd.


	For 0≤ k≤ n it holds that ^-∞_2k(U)^_H^+(U)≤ 2 while ^-∞_2k+1(U)^_H^+(U)=0. For every k, the spaces of R-even and R-odd invariants are each at most one-dimensional. 
	Moreover, any R-even invariant valuation is even, and any R-odd invariant valuation is odd. 

	In particular, any _H(U)-invariant valuation must be even.
Step 0. In the following, all forms are translation-invariant. For ξ∈ℙ_+(U^*), let ξ_⊥⊂ U be its annihilator. There is a natural identification of Ω_-∞^j,2n+1-j(U×ℙ_+(U^*))^tr with the generalized sections of the bundle over ℙ_+(U^*) with fiber ∧^j U^*⊗∧^2n+1-jξ_⊥⊗ξ^2n+1-j over ξ.
A Legendrian form corresponds to a section of the subbundle ξ⊗∧ ^j-1(U^*/ξ)⊗∧^2n+1-jξ_⊥⊗ξ^2n+1-j, which we then call a Legendrian section. We will find the ^+_H(U)-invariant generalized Legendrian sections. Let ψ(ξ) be such a section. 

There are three orbits under _H^+(U): the open orbit X_o={ξ≠ H^⊥} and two closed orbits X_c^±={± H^⊥}.

Step 1. Let us first show that ψ vanishes when restricted to the open orbit. Note that ψ|_X_o is smooth by _H^+(U)-invariance, and fix ξ∈ X_o.

Consider the stabilizer G_ξ:=^+(ξ)⊂_H^+(U), and G_ξ^1={g∈ G_ξ: g|_ξ=1}. Then ψ(ξ) is a G_ξ-invariant element in ∧ ^j-1(U^*/ξ)⊗∧^2n+1-jξ_⊥⊗ξ^2n+2-j.
By considering the various invariant subquotients, we deuduce the existence of an invariant element in one of the following spaces:

    V_1   =(H^⊥⊕ξ)/ξ⊗∧^j-2 (U^*/(ξ⊕ H^⊥))⊗∧^2n+1-j (ξ_⊥∩ H)⊗ξ^2n+2-j
    V_2   = (H^⊥⊕ξ)/ξ⊗∧^j-2 (U^*/(ξ⊕ H^⊥))⊗∧^2n-j(ξ_⊥∩ H)⊗ (ξ_⊥/ξ_⊥∩ H)⊗ξ^2n+2-j
    V_3   =  ∧^j-1 (U^*/(ξ⊕ H^⊥))⊗∧^2n+1-j (ξ_⊥∩ H)⊗ξ^2n+2-j
    V_4   = ∧^j-1 (U^*/(ξ⊕ H^⊥))⊗∧^2n-j(ξ_⊥∩ H)⊗ (ξ_⊥/ξ_⊥∩ H)⊗ξ^2n+2-j


Now take g_λ∈ G_ξ such that g_λ|_H=λ. Since g_λ has a 2n-dimensional eigenspace of eigenvalue λ on U, the action of g_λ on U^*, which is by (g_λ^-1)^*, has a 2n-dimensional invariant subspace of eigenvalue λ^-1. In the following we will simply write g_λ for this action. We may choose g_λ such that  g_λ|_ξ=λ^-1, g_λ|_H^⊥=λ^-2. Since ξ_⊥/(ξ_⊥∩ H)≃ U/H, g_λ acts on it by λ^2.

The action of g_λ on V_i is as follows: g_λ|_V_1=λ^-2-(j-2)+2n+1-j-(2n+2-j)=λ^-j-1, 
g_λ|_V_2=λ^-j, g_λ|_V_3=λ^-(j-1)+2n+1-j-(2n+2-j)=λ^-j, g_λ|_V_4=λ^-j+1.

Since j≥1, we conclude that a G_ξ-invariant can only exist in V_4, and only if j=1. However, ∧^2nξ_⊥⊗ξ^2n+1 has no  G_ξ-invariants: one may choose an element g∈ G_ξ with g|_ξ=1, while g≠ 1. Then (g:ξ_⊥→ξ_⊥) ≠ 1, and thus no such invariant exists. 

Step 2. We conclude that ψ is supported on X_c:={± H^⊥}. Since - commutes with _H^+(U), it acts on any space of _H^+(U)-invariants, which thus decomposes into a sum of even and odd invariants.
Now (-)=(-1)^2n+1=-1, so that odd closed Legendrian forms correspond to even valuations and vice versa. 

Let us show that for odd j there are no _H^+(U)-invariant closed Legendrian forms. Otherwise by what we proved, there is an _H^+(U)-invariant such form T supported on ξ=H^⊥ with a fixed orientation. Then T_s:=T-(-)^*T is odd, closed, Legendrian and _H^+(U)-invariant. Thus T_s defines an even _H^+(U)-invariant generalized valuation which is j-homogeneous. This contradicts Proposition <ref>.

Step 3.
The principal symbol of ψ, denoted F^α_X_c|_ξ  (see Appendix <ref>) over each point ξ∈ X_c  is an element of 
    W   := H^⊥⊗∧^j-1H^* ⊗∧ ^2n+1-jH⊗ (H^⊥)^2n+1-j⊗^*(T_ξℙ_+(U^*))⊗^α(T_ξℙ_+(U^*)) 
       =∧^j-1H^*⊗∧ ^2n+1-jH⊗ (H^⊥)^2n+2-j-α⊗(H)⊗(H^⊥)^2n⊗^α(H^*) .


Take δ_λ∈Scal_H(U) acting by λ∈ on H and by λ^2 on some fixed vector w∈ U∖ H. Then δ_λ acts on H^⊥ by λ^-2, and


    δ_λ|_W= λ^-(j-1)λ^2n+1-jλ^-2(2n+2-j-α)λ^-2nλ^4nλ^-α=λ^α-2
 hence α=2 is necessary for an invariant to exist. 

We may identify ∧^2n+1-jH≃∧^j-1H^*⊗∧^2nH. Thus 

    W= ∧^j-1H^*⊗∧^j-1H^*⊗^2(H^*) ⊗∧^2nH⊗ (H^⊥)^2n-j⊗(H)⊗(H^⊥)^2n

Let us find all w∈  W that are invariant under ^1_H(U). We may fix an ^1_H(U)-invariant symplectic form ω_H on H, so that ^1_H(U)-equivariantly, 

    W≃∧ ^j-1H^*⊗∧ ^j-1H^*⊗^2 (H^*).


By the fundamental theorem of invariant theory, an invariant element of W is given by fixing pairings of all the factors using ω, and then symmetrizing/antisymmetrizing accoridngly (some pairings could give zero). There would be j pairings. 

Note that R^*ω_H=-ω_H. Hence R|_H=(-1)^n and R=(-1)^n+1. We see that Rw=(-1)^j(-1)^n(-1)^j=(-1)^nw for any w∈ W^^1_H(U).

Step 4.
Now take ϕ∈^-∞_j(U)^_H^+(U) which is either even or odd, and denote T=T(ϕ).
Assume first that ϕ is R-even, that is _H(U)-invariant.
Then R^*T=· T. Thus its principal symbol is σ(T)(+H^⊥, -H^⊥)=(w_0, -w_0) for some w_0∈ W^^1_H(U). 
Noting that - acts by (-1)^j on W, it follows that ϕ is an even valuation if and only if j is even, which we may assume by step 2. By Proposition <ref>, _j^-∞(U)^_H(U)≤ 1.

Similarly if ϕ is R-odd, its principal symbol is σ(T)=(w_0, w_0) for some w_0∈ W^^1_H(U), and since j is even, ϕ must be odd. It remains to prove the dimension of the space of R-odd, j-homogeneous valuations is at most 1. It suffices to show by Theorem <ref> that the space of _H^+(U)-invariant, R-odd, j-homogeneous, closed Legendrian generalized forms supported on X_c is at most 1-dimensional. Any such form has the form T_+-R^*T_+ with T_+ supported on +H^⊥ and _H^+(U)-invariant. Clearly T_+-R^*T_+↦ T_++R^*T_+ is a bijection onto the corresponding R-even forms, which in turn bijectively correspond to _H(U)-invariant valuations. The R-even case now concludes the proof.





The following proposition completes the proof of Theorems <ref> and <ref>. 
 

For 0≤ k≤ n, it holds that ^-∞_2k(U)^_H(U)=1 while ^-∞_2k(U)^_H^+(U)=2.

The first assertion follows from the general construction of valuations on DH manifolds, as follows. By Proposition <ref>, we have at least n+1 linearly independent valuations ϕ_2j∈𝒲 _2j^-∞(U) which are invariant under the symmetries of U as a DH-manifold, in particular under translations and _H(U). Now Proposition <ref> concludes the proof.

The second assertion follows from the first one: by the last paragraph of the proof of Proposition <ref>, there is a bijection between the _H(U)-invariant valuations and the R-odd _H^+(U)-invariant valuations.




	1. Alternatively, one may use the Corfton valuations ψ_2j on the contact sphere constructed in section <ref> to prove the statement, by considering the translation-invariant valuations they define on the tangent spaces by Theorem <ref>.
	
	
	2. Examining the construction of ϕ_2k for general DH manifolds, we immediately see that ϕ^U_2k is in fact 2k-homogeneous.



	Let us describe explicitly the valuation ϕ_2n. Consider U=^2n+1 with the standard Euclidean structure, and take H={x_2n+1=0}. For u,v∈ H we let ω(u,v)=∑_j=1^n (u_2j-1v_2j-u_2jv_2j-1) be the standard symplectic form.
	Identifying U/H=H^⊥= e_2n+1= we recover the DH structure on U.
	
	Examining the proof of Proposition <ref>, we see that for convex K∈𝒦(V), ϕ_2n(K)=1/2(σ_K(H^⊥)+σ_K(-H^⊥)), where σ_Kdθ∈ℳ(S^2n) stands for the surface area measure of K, which is essentially just the push-forward of the 2n-dimensional Hausdorff measure on ∂ K to S^2n, see <cit.> for exact definition.
	
	




 §.§ Homogeneity 2n

The valuation of homogeneity 2n in U, which is just the surface area measure at the points of tangency to H⊂ U, has also a different natural setting, valid in dimension of any parity.


Let H^m⊂ V^m+1 be a hyperplane, and let _H, _V be fixed volume forms on H, resp. V. Consider _H(V):={g∈(V): g(H)=H,g|_H∈(H)}. Fix a Euclidean product P on V inducing the given Lebesgue measures on V and H. Assume for simplicity m≥ 2.
	
		ϕ_m(K):=1/2(σ_K(H^⊥)+σ_K(-H^⊥)) is the unique m-homogeneous, even _H(V)-invariant generalized valuation.
		
The Klain section of ϕ_m is a delta-measure on _m(V) supported on H. It is given by an element of (H)⊗^*(T_H_m(V))=(H)^m+2⊗^*(V)^m. Clearly _H(V) acts trivially on this space.

The proof of uniqueness is similar to Proposition <ref>. We first note that there are no invariant sections of the Klain bundle over the open orbit {E: E≠ H}⊂_m(V), since for such E there is a natural isomorphism E/E∩ H=V/H, so a (E)-invariant density on E would produce a (E)-invariant density on E∩ H, which clearly does not exist since (E) can rescale this space.
Next we use Lemma <ref> to study Klain sections supported on the closed orbit {H}: for α≥ 0, 
    F^α_{H}=(H)⊗^*(H^*⊗ V/H)⊗^α(H^*⊗ V/H)
 which is _H(V)-equivariantly isomorphic to ^α(H^*). There are no invariant polynomials on H, so we must have α=0, which readily yields a one-dimensional space of invariants. 
Next, we write an invariant Crofton formula for this valuation.
	
	There is an invariant Crofton measure μ_ over _1(V) which defines ϕ_m.  
	



Let θ(E)∈ [0,π] be the Euclidean angle between E∈^+_1(V) and H^⊥ with fixed orientation. Define the meromorphic family of functions f_s(E)=(cosθ)^*|t|^s where |t|^s∈ C^-∞() is the standard meromorphic family of even homogeneous generalized functions. Note that cosθ is submersive whenever t=cosθ=0, so the pull-back is well-defined. Since f_s(E) is invariant to orientation reversal, we get a generalized family on _1(V), still denoted f_s(E).

Let us identify translation-invariant measures (distributions) on _1(V) with (generalized) sections of the Crofton bundle _m over _1(V), whose fiber over E∈_1(V) is (V/E)⊗(T_E_1(V)).

Define a generalized section of _m over _1(V)∖_1(H) by μ_:=f_-m-2(E) when m is even, and μ_:=_s=-m-2f_s when m is odd.

Let us check μ_ is _H(V)-invariant. 
For g∈(V) consider the Jacobian ψ_g(E)=Jac(g:E→ gE)^-2, where E, gE are endowed with the volume induced by the Euclidean product P. It follows that for g∈_H(V) and E⊄H, 
    ψ_g(E)=cos^2 ∠ (g E, H^⊥)/cos^2 ∠ (E, H^⊥).
  Now f(E) represents (with respect to the Euclidean trivialization) an _H(V)-invariant Crofton measure precisely if   
g^*f(E)=ψ_g(E)^-m+2/2f(E). Since g^* f_s=ψ_g(E)^s/2 f_s(E), we conclude μ_ is indeed _H(V)-invariant.

It remains to verify these Crofton measures define non-zero valuations. For this, we evaluate ϕ_m(B) for the unit ball B^m+1.
By definition, ϕ_m(B)=1. On the other hand, writing ω_m for the volume of the Euclidean ball B^m, we get 


    (f_s(E)dE)(B)= ω_m ∫__1(V)  |cosθ(E)|^s dE=ω_m  ∫_0^π |cosθ|^ssin^m-1θ dθ/∫_0^πsin^m-1θ dθ
       =ω_m   ∫_0^1 t^s (1-t^2)^m-2/2dt /∫_0^1(1-t^2)^m-2/2dt=ω_m ∫_0^1 u^s-1/2 (1-u)^m-2/2du /∫_0^1 u^-1/2 (1-u)^m-2/2du =ω_mB(s+1/2, m/2)/B(1/2, m/2)

Thus 
    C_m=π^m/2/Γ(1+m/2)Γ(s+1/2)/Γ(s+1/2+m/2)Γ(m/2+1/2) /Γ(1/2)=π^m/2-1/2/Γ(1+m/2)Γ(s+1/2)Γ(m/2+1/2)/Γ(s+1/2+m/2)

When m=2n is even, s=-2n-2 and  
    C_2n=-1/2π^n-1/n!Γ(n+1/2)Γ(-n-1/2)

When m=2n-1 is odd, s=-2n-1, Γ(1+s/2) has a simple pole with residue 2_z=-nΓ(z)=2(-1)^n1/n!, and 

    C_2n-1=2(-1)^n1/n!π^n-1/Γ(n+1/2)Γ(n)/Γ(-1/2)=(-1)^n-1π^n-3/2/nΓ(n+1/2)
 



We can now recover effortlessly the inverse form of the Koldobsky-Ryabogin-Zvavitch formula  <cit.>.

	Let K⊂^m+1 be a smooth, symmetric convex body, and let κ(x) be its Gaussian curvature and ν(x) the unit normal at x∈∂ K. Then there is a universal explicit constant C_m (computed in the proof of Proposition <ref>) s.t. for even m=2n, 
    κ(x)^-1=C_m ^-1∫_S^m_m(_θ^⊥(K))dσ_1(θ)/⟨θ,ν(x)⟩^m+2
 
	while for odd m, 
	
    κ(x)^-1   =C_m^-1⟨δ^(m+1)(⟨ν(x), θ⟩),_m(_θ^⊥(K))⟩
       =2/(m+1)! C_m^-1.d^m+1/dt^m+1|_t=0((1-t^2)^m-2/2∫_θ:⟨θ, ν(x)⟩=cos t_m(Pr_θ^⊥(K))dσ_1(θ))

	


Since _m(_η^⊥(K)) is the cosine transform of σ_K(θ)dσ_1(θ), one can think of this formula as simply the inversion of the cosine transform. The integral can be viewed as the (-2n-2)-cosine transform of the support function of the projection body.





§ CONTACT MANIFOLDS



 §.§ Specializing from general DH manifolds.

Recall <ref> that a contact manifold has a canonical structure of a DH manifold. Let us recall a few basic facts.

The standard contact structure on ^2n+1={(x_1,…, x_n, y_1, …, y_n,z)} is given by dz=∑_j=1^n x_jdy_j-y_jdx_j.
Its symmetry group contains translations along the z-axis and rotations in the (x_j, y_j) planes.

[Darboux-Pfaff] Take a contact manifold M^2n+1. Then any x∈ M has a neighborhood contactomorphic to an open subset of the standard contact ^2n+1.


We will make use of the following simple observation.
 
	In the standard contact space ^2n+1, it holds that a closed, smooth, strictly convex hypersurface F⊂^2n+1 is normally transversal.

We will use the standard Euclidean structure to identify ℙ_^2n+1 with ^2n+1× S^2n.
Let p be a contact point of F. We may assume p=(a_1,0,…, a_n,0,0). As T_pF=H_p={dz=∑ a_j dy_j}, (x_j, y_j)_j=1^n form a system of coordinates for F near p, and so locally F is the graph z=f(x_1,…, y_n) of a strictly convex or strictly concave function f. We will denote w=w(x)=(x_1,…, y_n), x=(w,f(w)).
Let ν_w=ν_x be the normal to F at x, and h_w=h_x the normal to H_x, both normalized to have z coordinate 1 (the other pair of normals has z=-1 and is treated identically). Thus ν_w=(-f'_x_1,…, -f'_y_n, 1), and h_w=(y_1, -x_1,…, y_n, -x_n, 1). 

Claim. There exists c=c(f,p)>0 such that ν_w-h_w≥ cw-w(p) in a neighborhood of p.

Proof. Write 

    E=ν_w-h_w= ∑( x_j-f'_y_j)^2+ (y_j+f'_x_j)^2

Replacing f with f-∑ a_j y_j and x_j with x_j-a_j, E remains unchanged, and the convexity of f is retained. We thus may assume p=0, and ∇ f(p)=0. Hence

    E=w^2+∇ f(w)^2-2∑_j (x_jf'_y_j(w)-y_jf'_x_j(w))=w^2+∇ f(w)^2-2ω(w,∇ f(w))

where ω(w,w')=∑ (x_jy'_j-x'_jy_j) is the standard symplectic form on ^2n. We clearly may replace f with its quadratic approximation at p, namely f(w)=1/2⟨ Aw, w⟩ for A=Hessf(p). Then ∇ f(w)=Aw, and by Writinger's inequality |ω(w,∇ f(w))|≤wA wsin∠ (w, Aw). We conclude that 

    E   ≥w^2+Aw^2-2wAwsin∠(w,Aw)
       =(w-Aw)^2+2wAw(1-sin∠(w,Aw))

Since A is sign-definite, it holds that λ_minw≤Aw≤λ_maxw and |⟨ Aw,w⟩|≥λ_minw^2,  
where λ_min/max is the minimal/maximal eigenvalue of |A|. It follows that |cos∠ (w, Aw)|=|⟨ w, Aw⟩|/wAw≥λ_min/λ_max, so 1-sin∠(w,Aw)>1-√(1-λ_min^2/λ_max^2)=:c_0, and hence
E ≥ 2c_0λ_minw^2 as claimed.
Recall M_H={(x, h_x):x∈^2n+1} is 2n+1 dimensional, and NF={(x,ν_x): x∈ F} has dimension 2n. Thus we ought to show T_(p, ν_p)M_H∩ T_(p, ν_p)NF={0}. If not, there is a curve x(t)⊂ F, x(0)=p with non-zero velocity vector, such that h_x(t) and ν_x(t) have equal velocity vectors at t=0. But then we should have h_x(t)-ν_x(t)=O(t^2) as t→ 0, contradicting the claim. 

 
	Assume M^2n+1 is a contact manifold. Then ϕ_2j, 0≤ j≤ n are all linearly independent, while ϕ_2j-1 is a linear combination of ϕ_2i, j≤ i≤ n.

The first statement is just part i) of Proposition <ref>. For the second statement, we will use the same notation as in the proof of Proposition <ref>.

Write h=(h_ij)_i,j=1^2n=H_s+H_a, the symmetric and antisymmetric part.

First, we note that in a contact manifold, the relation between the symplectic and horizontal structures is given by H_a=-J. Indeed, α(v)=g(R, v) so we have


    dα(u,v)   =L_uα(v)-L_vα (u)-α([u,v])=L_u g(R,v)-L_vg(R, u)-α([u,v])
       =g(∇ _u R, v)+g(R, ∇_u v)-g(∇_v R, u)-g(R, ∇_v u)-g(R, [u,v])
       =g(∇_u R, v)-g(∇_v R, u).



Hence h_ji-h_ij=θ_j(∇_X_iR)-θ_i(∇_X_jR)=dα(X_i, X_j)=J_ij.

Take F a closed hypersurface which is normally transversal. Introduce the notation Ψ_k(F, p):=2n  k^-1| A_p| ϕ_2n-k(F, p). Then


    Ψ_k(F, p)   =D(H-S[k], J[2n-k])=(-1)^kD(S-H_s+J[k], J[2n-k]
       = (-1)^k∑_i=0^kk iD(S-H_s[i], J[2n-i])

Observe that for 2n× 2n matrices X, Y for which X^T=X, Y^T=-Y it holds that D(X[2n-i], Y[i])=0 for odd i. We conclude  

    Ψ_k(F, p)= (-1)^k∑_i=0^⌊k/2⌋k 2iD(S-H_s[2i], J[2n-2i])

that is,

    ϕ_k(F,p)=(-1)^k2n  k| A_p|^-1∑_i=0^⌊2n-k/2⌋2n-k 2iD(S-H_s[2i], J[2n-2i])

Thus for normally transversal closed hypersurface F, ϕ_2j-1 is a fixed linear combination of the ϕ_2i with j≤ i≤ n.
By Darboux-Pfaff, we may cover M by charts U_α that are  contactomorphic to open subsets  V_α of the standard contact space ^2n+1. By Lemma <ref>, a closed, smooth, strictly convex hypersurface in V_α is normally transversal to the contact structure. The statement now follows from Lemma <ref>, Proposition <ref>, and Lemma <ref>.





We now prove a Hadwiger theorem for contact manifolds.


	Let M^2n+1 be a contact manifold. Then 𝒱^-∞(M)^Cont(M) is spanned by ϕ_2k, 0≤ k≤ n.



For an element ϕ∈𝒲_k^-∞, we will write [ϕ]_k for its image in 𝒲_k^-∞/𝒲_k+1^-∞.
We will show by induction on 2n+1-k that 𝒲_k^-∞(M)^Cont(M) is spanned by (ϕ_2j)_2j≥ k. 
Take ϕ∈𝒲_2n+1^-∞(M)^Cont(M)=ℳ^-∞(M)^Cont(M)=ℳ^∞(M)^Cont(M),  where the latter equality holds since Cont(M) acts transitively on M, so that an invariant distribution is automatically smooth.
There is no such invariant measure, hence ϕ=0. 

Assume now that  ϕ∈𝒲_2n+1-k^-∞(M)^Cont(M). 
Recall 
    𝒲_2n+1-k^-∞(M)/𝒲_2n+2-k^-∞(M)=(𝒲_k,c^∞(M)/𝒲_k+1,c^∞(M))^*=Γ_c (M, _k^∞(TM))^*

Hence by Proposition <ref>, 
    [ϕ]_2n+1-k∈(Γ_c (M, _k^∞(TM))^*)^Cont(M)=Γ^∞(M, _2n+1-k^-∞(TM))^Cont(M)=
 where we used the Alesker-Poincaré isomorphism 
^∞_k(T_xM)^*=_2n+1-k^-∞(T_xM)⊗(T_xM).
For a point x∈ M, its stabilizer in Cont(M) is (x)=_H(T_x M). Hence 
Γ^∞(M, _2n+1-k^-∞(TM))^Cont(M)=_2n+1-k^-∞(T_xM)^_HT_xM.
By Proposition <ref>, the latter space of invariants is trivial if k is even, and one dimensional if k is odd. 

Thus if k is even, ϕ∈𝒲_2n+2-k^-∞(M). In case k is odd, we use part (i) of Proposition <ref> to find a multiple of ϕ^M_2n+1-k such that [ϕ]_2n+1-k=c[ϕ^M_2n+1-k]_2n+1-k⇒ϕ-cϕ^M_2n+1-k∈𝒲_2n+2-k^-∞(M)^Cont(M). The induction assumption now completes the proof.



	Unlike the contact case, we do not have a uniqueness result in the DH category, where the symmetry group is in general trivial. In the Riemannian setting, uniqueness of isometry-invariant valuation assignment can be deduced from the classical Hadwiger theorem in conjunction with the Nash embedding theorem. This last piece is missing in the DH setting. A different type of uniqueness in terms of the Cartan frame apparatus, which is again tailored to the Riemannian setting, was established by Fu-Wannerer <cit.>.
	




 §.§ A dynamical point of view


Let F⊂ M^2n+1 be a hypersurface, and let p∈ F be a normally transversal contact point: T_pF=H_p. Denote by F_H the singular hyperplane field on F given by F_H|_p=H_p∩ T_pF. When M=3, this field integrates to the characteristic foliation.

One can describe ϕ_2k(F, p) explicitly through the singular bundle H_x near x. 


Let β∈Ω^1(F) be a form defined near p such that β=F_H. Since M is contact, we may assume dβ≠ 0 near p (e.g. by taking β=α|_F for some contact form α on M), and there is a unique vector field B∈𝔛(F) near p such that i_Bdβ=β. 
In particular, B(p)=0 and B(x) is tangent to the characteristic foliation.
If β'=fβ is a different form with d_pβ'≠ 0 f(p)≠ 0, the corresponding vector field is 

    B' = (1+df(B)/f)^-1B

Since df(B)(p)=df(B(p))=0, the differential d_pB∈𝔤𝔩(T_pB) only depends on F_H. 


	Note  that d_pB determines whether p is an elliptic or hyperbolic singular point of the characteristic foliation.




	F is normally transversal at p if and only if d_pB is non-singular. In that event	

	
    ϕ_k(F, p) = | d_pB|^-1(∧ ^2n-k d_p B)


Again we use notation from the proof of Proposition <ref>.

Since all contact manifolds are locally isomorphic, we work in ^2n+1 with coordinates (x_1,…, x_n, y_1, …, y_n, z), and contact form α=-dz+∑ x_jdy_j. We may assume further that p=0 and T_pF={z=0}. Then (x_j, y_j) are local coordinates on F, and β=∑ -∂ f/∂ x_j dx_j + (x_j-∂ f/∂ y_j) dy_j, dβ|_0=∑ dx_j∧ dy_j. It follows that B(x,y)^T=(x,0)^T+J∇ f, where ∇ f= (∂ f/∂ x_j, ∂ f/∂ y_j)^T. Thus 
    d_0B= ([ I_n   0;   0   0 ]) +J H^2f


On the other hand, one immediately computes that 
    h= ([   0   0; I_n   0 ])
 and S=H^2f, hence h-S=Jd_0B. This readily shows that F is normally transversal at p, which is equivalent to the non-singularity of h-S, if and only if d_p B≠ 0; and 
    D(h-S[2n-k], J[k])= J· D(d_0B[2n-k], I_n[k])=2n  k^-1(∧^2n-k(d_0 B)).

By Definition <ref>, we are done.


Computatiton of ϕ_2k is straightforward with this approach. Here is a simple proof of a well-known fact.

In the standard contact space ^3 with contact structure given by dz=xdy-ydx, spheres of different radii are not equivalent through a contactomorphism of the ambient space.

One computes that ϕ_2(S_R)=8(1+1/4 R^-2)^-1.


[The contact sphere]Let us compute ϕ_2k(S^2m) in S^2n+1. By Theorem <ref>, we may assume n=m. Consider S^2m+1⊂^m+1, with coordinates x_1,y_1, …, x_m+1, y_m+1. The contact form is given by 
    α_p(v)=⟨ v, √(-1)p⟩=∑_j=1^m+1(-y_jdx_j+x_jdy_j).
 so that dα=2∑_j=1^m+1 dx_j∧ dy_j. 
Fix S^2m={y_m+1=0}. Then the two unique contact points of S^2m are given by x_m+1=±1, and we use the coordinates (x_j, y_j)_j=1^m near those points.
In those coordinates, β=α|_S^2m=∑_j=1^m(-y_jdx_j+x_jdy_j), and dβ=2∑_j=1^m dx_j∧ dy_j.
Then 

    i_Bdβ=β⇒ 2∑ (dx_j(B)dy_j-dy_j(B)dx_j)=∑ (x_jdy_j-y_j dx_j)

so that B(p)=1/2p. Thus d_0B=1/2 I_2m, and

    ϕ_2k (S^2m)= 2(∧^2m-2kI_2m)=22m  2k





§ SYMPLECTIC-INVARIANT DISTRIBUTIONS




 §.§ Linear algebra


The real anti-symmetric matrices of size 2N× 2N will be denoted _2N. 
Define SDiag(λ_1,…,λ_N)∈_2N() to be the block-diagonal matrix consisting of the 2× 2 blocks 

    SDiag(λ_j)=([    0  λ_j; -λ_j    0 ])

The following is a standard fact from linear algebra.

	For A∈_2N there is a matrix B∈_2N() and D=SDiag(λ_1,…,λ_N) such that A=B^TDB. The vector (λ_1,…,λ_N) is uniquely defined up to permutations and signs of the λ_j.
 
Denote Δ_N={λ_1≥…≥λ_N≥ 0}. For A∈_2N, let Λ(A)∈Δ_N be the unique vector such that A=B^T·SDiag(Λ(A))· B for some B∈_2N().

Recall the multi-Kähler angles 0≤θ_1≤…≤θ_κ≤π/2 of E∈^_2k(^n) introduced by Tasaki <cit.>, where κ=min(k, n-k). They are defined as follows: choose an orthonormal basis (e_i)_i=1^2k of E, and define the symplectic Gramm matrix A=ω(e_i, e_j). Then Λ(A)=(cosθ_1,…,cosθ_κ).

	Let θ_i, i=1,…,κ be the multi-Kähler angles of a subspace E∈^_2k(^n) chosen at random (with respect to the (2n)-invariant probability measure). Then the probability distribution of  (cosθ_i)_i=1^κ is uniform in Δ_κ.

We may assume 2k≤ n so that κ=k.
We first observe that the distribution is independent of n. Indeed we may condition on the event E⊂ F where F is any fixed complex k-dimensional subspace in ^n, but the distribution of the multi-Kähler angles is clearly independent of F. Thus we assume n=2k.

Next notice that for a symplectic subspace E, there is a unique, up to order, decomposition E=E_1⊕…⊕ E_k, where E_j=2 and all F_j:=ℂ E_j are pairwise orthogonal.
This decomposition can be found as follows: for an orthonormal basis e=(e_j)_j=1^2k of E, consider the matrix M(E, e)=(ω(e_i, e_j))_i,j=1^2k∈_2k. 
If (e') is a different orthonormal basis, there is an equality of row vectors (e'_i)=(e_j)B for some B∈(2k), and one checks that M(E, e')=B^TM(E, e)B. 
Thus for generic E there is a unique, up to order, orthonormal basis e such that M(E, e)=SDiag(Λ(M(E,e))). We then set E_j=(e_2j-1, e_2j) for 1≤ j≤ k.

For a given decomposition ^2k=F_1⊕…⊕ F_k into orthogonal copies of ^2, the multi-Kähler angles of E are the collection of Kähler angles of E∩ F_j∈_2^(F_j). Thus conditioning on the decomposition, we conclude the multi-Kähler angles of E are independent and identically distributed, and it remains to find the distribution of the Kähler angle of a real 2-plane E⊂^2. 

We will work with the oriented Grassmannian.
Consider ^4=^2 with 
    ω((x_1+ix_3, x_2+ix_4), (x'_1+ix'_3, x_2+ix'_4))=x_1x_3'-x_1'x_3+x_2x_4'-x_2'x_4.
 
Let (u,v) be an orthonormal basis for E∈^+_2(^4).
We identify _2^+(^4)=S^2× S^2, E↦ (z, w) using the standard Euclidean structure and the Plücker embedding ^+_2(^4)⊂ S(∧^2 ^4), E↦ u∧ v=(x_12,…, x_34) where x_kl=u_kv_l-u_lv_k, followed by the change of coordinates 


    x_12    :=w_1+z_1/2   x_34:=w_1-z_1/2   x_13:=-w_2+z_2/2
    
    	x_24    :=w_2-z_2/2   x_14:=w_3+z_3/2   x_23:=w_3-z_3/2.


The corresponding measure on S^2× S^2 is the standard one. Then for E=(z,w)∈ S^2× S^2, and u, v an oriented orthonormal basis of E, ω(u, v)=x_13+x_24=-z_2. Denoting by θ_E the Kähler angle, we conclude that cosθ_E=|Λ(E)|=|ω(u,v)|=|z_2| is distributed uniformly in [0,1] by the theorem of Archimedes.



Let Δ^1_κ={1≥λ_1≥…≥λ_κ≥ 0}.
Denote by Λ:_2k^(^n)→Δ^1_κ the vector (cosθ_i)_i=1^κ. 
We conclude that Λ_*(dσ_1)=κ!∏_i=1^κ dλ_i.



 §.§ Powers of the Pfaffian


Consider the meromorphic families of generalized functions ||_±^s∈ C^-∞(_2N()), s∈ℂ.
They can be constructed by first considering s>-1, whence ||_±^s is an integrable function, and then using the Cayley-type identity (see e.g. <cit.>)

    (∂) (X)^s+1= (s+1)(s+3)…(s+2N-1)(X)^s

for a meromorphic extension to s∈. Thus the poles of both families are at s=-1,-2,…. 

[Muro <cit.>]
	The linear combination |(X)|^s:=|(X)|_+^s+|(X)|_-^s is analytic at even s∈ℤ and has a simple pole at odd s≤ -1. The linear combination (X)|(X)|^s:=|(X)|_+^s-|(X)|_-^s is analytic at odd s∈ℤ and has a simple pole at even s≤ -2.




Let (V,ω) be a 2n-dimensional symplectic space, P a compatible Euclidean structure with corresponding complex structure J, so that ω(u,v)=P(Ju,v). Then P induces a Lebesgue measure _P(E) on all subspaces E⊂ V. Define σ_ω,P:^+_2k(V)→[-1,1] by σ_ω, P(E)=ω^k|_E/_P(E). We will often omit P from the index when no confusion can arise. 

We will now define meromorphic families of distributions on ^+_2k(^2n) given by μ_±(s)=|σ_ω(E)|_±^sdσ_1(E) for large (s). The construction is virtually identical to the one carried out in <cit.>, with (2n) replacing (p,q). We present it here for the reader's convenience.

We will assume for now that 2k≤ n. 

Let U⊂_2k(V) be an open set. Let B_E=(u_1(E),…,u_2k(E)):U→ V^k be a smooth field of P-orthonormal bases of E∈ U. Define the function M_P:U→_2k(ℝ) given by M_P(E)=ω(u_i(E),u_j(E))_i,j=1^2k. Note that σ_ω(E)= M_P(E).

Denote by U_P⊂_k(V) the open, dense subset of subspaces E∈_2k(V) for which E∩ JE={0}. Clearly E∈ U_P if and only if 1∉Λ(M_P(E)).


	M_P is a proper submersion at every E∈ U∩ U_P.


Consider a curve γ_1 through E given by 
    γ_1(t)=(u_1(t),u_2,…,u_2k)
 with u_2,…, u_2k fixed, and ξ=u̇_1(0)∈ E^P arbitrary. It follows that 

    D_E M_P (γ̇_̇1̇(0))=
    [          0  ω(ξ, u_2)          ⋯  ω(ξ,u_2k);  -ω(ξ,u_2)          0          ⋯          0;          ⋮          ⋮          ⋱          ⋮; -ω(ξ,u_2k)          0          ⋯          0 ]

Since E∈ U_P, ω(u_j, ∙)_j=2^2k are linearly independent functionals in ξ∈ E^P, and so the first row of D_Λ M_P(γ̇_̇1̇) is arbitrary, while the other entries in the upper triangle vanish. Replacing  γ_1 with γ_j in the obvious way, we conclude D_E M_P(∑α_j γ̇_j(0)) can be arbitrary, thus concluding the proof.



	One can choose finitely many ω-compatible Euclidean structures P_i s.t. {U_P_i} cover _2k(V).

Given E∈_2k(V), a generic choice of an ω-compatible (P, J) would have E∩ JE={0} by a trivial dimension count. Fixing one such J=J(E) with corresponding P(E) for every E, we get an open cover (U_P(E))_E∈_2k(V) of _2k(V). The claim now  follows by the compactness of _2k(V).



We now explain how to pull-back |(X)|^s_± to _2k^+(V), using the locally-defined submersion M_P.

	For s∈ℂ, let 𝒟^s be the line bundle of s-densities over _2k^+(V), which has fiber ^s(E) over E∈_2k^+(V). 
	We say that a choice of generalized section f(s)∈Γ^-∞(U, 𝒟^s) over U⊂_2k^+(V) for s∈Ω⊂ is meromorphic in s if, having fixed a Euclidean metric P and using it to identify all bundles 𝒟^s, one obtains a map f_P:Ω→  C^-∞(U) which is meromorphic in s. 
	
	We denote by 𝔐^-∞(𝒟^s) the sheaf for which Γ(U, 𝔐^-∞(𝒟^s)) is the space of meromorphic in s maps ℂ→Γ^-∞(U, 𝒟^s). 
	



Recall the orbits X^2k_±,  (X^2k,+_r)_r=0^κ-1  of ^+_2k(V) under (V) defined in the paragraph following eq. (<ref>), where κ=min(k,n-k). In terms of multi-Kähler angles, E∈ X^2k,+_r precisely when exactly r of the angles are distinct from π/2.

We are now ready to construct the meromorphic families.

	There are global sections f_±(s)=M_P^*||^s_± of 𝔐^-∞(𝒟^s) supported on X^2k_±, s.t. whenever s is not a pole of f_±,  f_±(s) is (V)-invariant.

Assume first 2k≤ n. Let P_i be a finite collection of ω-compatible Euclidean structures as in Lemma <ref>, and let U_i=U_P_i⊂^+_2k(V) be the corresponding open sets of generic subspaces. For each i, cover U_i by open sets U_ij⊂ U_i so that M_ij=M_P_i:U_ij→_2k can be defined by some smooth field of orthonormal bases of E over U_ij. Now since M_ij is a proper submersion, one obtains a meromorphic in s family of functions f̃^±_ij(E;  s)∈ C^-∞ (U_ij) given by f̃_ij^±(∙; s)=M_ij^*||_±^s.

It then obviously holds that on U_ij∩ U_ij', f̃^±_ij(Λ; s) and f̃_ij'^±(Λ; s) coincide as continuous functions for (s)>0. Therefore, they coincide on U_ij∩ U_ij' as meromorphic functions, and we may merge all f̃_ij^± into one meromorphic family f̃_i^±(∙; s)∈ C^-∞ (U_i). The corresponding (through P_i) section f_i^±∈Γ(U_i, 𝔐^-∞(D^s)) is obviously (V)∩(P_i)-invariant. Moreover, it is 𝔰𝔭(V)-invariant.

Next, we claim that f_i^± and f_i'^± coincide on U_i∩ U_i'. Since both are meromorphic, we may assume in the following that (s)>0.

It is easy to see, using Lemma <ref> as in the proof of Proposition <ref>, that for (s)>0, no (V)-invariant generalized sections of 𝒟^s can be supported on a set of positive codimension: 
using Lemma <ref>, we consider for α≥0 the bundle F^α_X^2k, +_r over X^2k,+_r where r>0, that has fiber (E)^s⊗^*(N_EX^2k,+_r)⊗^α (N_EX^2k,+_r) over E.
Denoting E_0=E∩ E^ω, by Corollary <ref>

    F^α_X^2k,+_r|_E=(E)^s⊗(∧^2E_0)⊗^α (∧^2E_0^*)

which clearly contains no (V)-invariants.  It follows that the space of (V)-invariants in Γ^-∞(^+_2k(V), 𝒟^s) supported on X^2k_ϵ is at most 1-dimensional for each ϵ∈{±}. 

Since U_i⊂^+_2k(V) is dense, it follows by construction that for (s)>0, f^±_i(∙; s) extends by continuity to an (V)-invariant section of 𝒟^s over ^+_2k(V) supported on X^2k_±, and by the previous paragraph we can find meromorphic functions c_i(s), such that c_1(s)=1 and the sections c_i(s) f_i^±(Λ;  s) coincide for all i. Denoting by p_i(s)∈Γ^∞ (^+_2k(V), 𝒟^s) the Euclidean section defined by P_i,  it holds that 
    f^±_i(E;s)= | M_P_i(E)|^sp_i(s)
 for E∈ X^2k_±, so that 
    | M_P_1(E)|^sp_1(s)=c_i(s) | M_P_i(E)|^sp_i(s)

implying 

    c_i(s)=(| M_P_1(E)|/| M_P_i(E)|p_1(1)/p_i(1))^s
 
for all E∈ X^2k_±. Since c_i(s) is independent of E, one has c_i(s)=c_i^s for some c_i>0. 

Finally, for s=1, 𝒟^1 is the Klain bundle of Lebesgue measures, and it is easy to see that all f_i^±(1) represent the Liouville measure induced by the symplectic form on every symplectic subspace E.  It follows that c_i=1⇒ c_i(s)≡ 1. Thus we have shown that f_i^±, f_i'^± coincide in Γ(U_i∩ U_i', 𝔐^-∞(𝒟^s)).

We conclude there is a globally defined section f_± of 𝔐^-∞(𝒟^s) which is 𝔰𝔭(V)-invariant and supported on X^2k_±, respectively. For (V)-invariance, we observe it holds for (s)>0 and then invoke uniqueness of meromorphic continuation. 

This concludes the proof when 2k≤ n. For the case 2k>n, we simply use the oriented skew-orthogonal complement map ^+_2k(V)→^+_2n-2k(V) to pull-back f_s. This is a valid operation since we have the equivariant identification E^ω≃ (V/E)^*⇒(E^ω)≃(E).





	Set |σ_ω|_±^s∈ C^-∞(_2k^+(V)) to be the value of f_±(s) under the Euclidean trivialization.


Write κ=min(k,n-k).
One has ∫__2k(^2n)|σ_ω|_±^s dσ_1(E)=1/2κ!(s+1)^κ.

Using the cosines of the multi-Kähler angles in decreasing order, denoted Λ(E)=(λ_1,…, λ_κ), λ_j=cosθ_j, we have |σ_ω(E)|=∏_j=1^κλ_j. Then for (s)>0, using Proposition <ref> we get

    ∫__2k(^2n)|σ_ω|^s dσ_1(E)= 1/κ!∫_[0,1]^κ∏_j=1^κλ_j^sdλ_j  = 1/κ!(s+1)^κ

and the result follows by uniqueness of meromorphic extension.


The two values s=-2n,-(2n+1) are of particular interest, as evidenced in the two theorems below. The first theorem concerns the linear Grassmannian:


	The distribution μ^+_ω∈ℳ^-∞(^+_2k(V)) given by μ^+_ω:=|σ_ω(E)|^-2ndσ_1(E) is (V)-invariant, has full support and ∫__2k(V)μ_ω^+≠ 0. It is even with respect to orientation reversal.

In particular, we get a canonically normalized (V)-invariant distribution μ_ω:=π_*μ_ω^+ on _2k(2n), where π:^+_2k(V)→_2k(V) is the double cover map.
A distribution over ^+_2k(V) is a generalized section of the bundle with fiber (T_E_2k^+(V)) over E, which is (V)-isomorphic to 𝒟^-2n. All statements follow immediately from Proposition <ref>, Theorem <ref> and Lemma <ref>.
We conjecture that μ_ω is the unique (V)-invariant distribution on _2k(V). This was shown by Gourevitch, Sahi and Sayag in <cit.> for k=n when n is even.

Similarly, we have a statement for the affine Grassmannian. We define |σ_ω(E)|^s∈ C^-∞(^+_2k(V))^tr by pulling-back by the projection map ^+_2k(V)→^+_2k(V). Let dE be the measure on _2k^+(V) which is built out of dσ_1 on _2k^+(V), and the Euclidean measure on translations. Define the odd distribution 
    μ_ω:=(σ_ω)|σ_ω(E)|^-2n-1dE∈ℳ^-∞(^+_2k(V))
 and the even distribution  
    μ_0:=_s=-2n-1|σ_ω(E)|_+^sdE∈ℳ^-∞(^+_2k(V)).




The distributions μ_ω, μ_0 are (V)-invariant. μ_ω has full support. 

The proof is as in the linear case.
In particular, there is a an even, canonically normalized (V)-invariant distribution μ_0:=π_*μ_0^+ on _2k(V), supported on the ω-degenerate subspaces.


We will need the following a-priori information about the wavefront set of μ_ω.


	The wavefront set of μ_ω belongs to ∪_rN^*X^2k_r.

This is immediate from (V)-invariance.




§ THE CONTACT SPHERE






 §.§ A Crofton basis for S^2n+1


In this section, V=^2n+2, M=S^2n+1=ℙ_+(V). Take ξ∈ℙ_+(V) so that T_ξ S^2n+1=ξ ^*⊗ V/ξ. The contact hyperplane is H_ξ=ξ^*⊗ξ^ω/ξ⊂ T_ξℙ_+(V). Then T_ξ  S^2n+1/H_ξ=ξ^*⊗ V/ξ^ω≃(ξ^*)^⊗ 2 (ξ)-equivariantly. The symplectic form on ξ^ω/ξ defines a form 
ω_ξ∈∧^2 H_ξ^*⊗ (ξ^*)^⊗ 2=∧^2 H_ξ^*⊗ T_ξ  S^2n+1/H_ξ.
We remark that the form ω_ξ is determined by the contact distribution alone, without reference to the form ω on V, see Example <ref>.
The linear symplectic group (V) acts on S^2n+1 by contactomorphisms. The stabilizer of ξ∈ S^2n+1 is easily seen to act on T_ξ S^2n+1, which is a dual Heisenberg algebra, by its full group _H(2n+1) of automorphism.

Now (2n+2)-_H(2n+1)=(n+1)(2n+3)-(n+1)(2n+1)=2n+2= S^2n+1+1, so that (ξ)=_H(2n+1)+1, and we denote (ξ)=^*_H(2n+1). The kernel of the restriction homomorphism  ^*_H(2n+1)→_H(2n+1) consists of the linear maps that fix all v∈ξ^ω and acts on some w∈ V∖ξ^ω by w↦ w+λξ, for some λ∈.

Thus S^2n+1=(2n+2)/_H^*(2n+1). Inspired by the analogy to the Riemannian symmetric space presentation S^2n+1=(2n+2)/(2n+1), we look for Crofton-type formulas for the contact valuations on the sphere.

Consider the double fibration 

    W[dl]_τ[dr]^π   
    _2k(V)        S^2n+1
 
where W is the partial flag manifold {(E, θ)∈_2k(V)× S^2n+1: θ∈ E }



	
	For 0≤ k≤ n, define the generalized valuation ψ_2k through the Crofton formula (in the sense of subsection <ref>) 
    ψ_2k:=π_*τ^*(μ_ω)=∫__2n+2-2k(V)χ(∙∩ E)σ_ω(E)^-2n-2dσ_1(E)∈𝒱^-∞(S^2n+1)^(V)
 




It follows from Lemma <ref> that ψ_2k∈𝒲^-∞_2k(S^2n+1). Moreover, since ψ_2k(S^2k)≠ 0 by Theorem <ref> (see the computation preceding eq. (<ref>) below for precise value), we conclude ψ_2k∈𝒲^-∞_2k(S^2n+1)∖𝒲^-∞_2k+1(S^2n+1), and in particular all ψ_2k are linearly independent.


	𝒱^-∞(S^2n+1)^(V) is spanned by χ,ψ_2,…,ψ_2n.

The proof is identical to that of Theorem <ref>, with (V) replacing the full group of contactomorpisms. The proof remains valid since (V) acts transitively, with the same action of the stabilizer on the tangent space by _H(2n+1).
This completes the proof of Theorem <ref>.
In light of Theorem <ref> we get
 
 	For 0≤ k≤ n, ψ_2k are linear combinations of ϕ_2j, 0≤ j≤ n. In particular, ψ_2k is invariant under all contactomorphisms of ℙ_+(V). 
 
Thus we establish Theorem <ref>, except for the explicit determination of the coefficients which is deferred to the next subsection.

It follows also that ψ_2k(F) is well-defined for subsets F⊂ S^2n+1 normally transversal to the contact distribution. We will make use of the following lemma.


	Assume that a closed submanifold F⊂ S^2n+1 is normally transversal, and χ_F:=χ(F∩∙)∈ C^-∞(_2n+2-k(V)) has wavefront disjoint from (μ_ω). Then
	
    ψ_2k(F)=∫__2n+2-2k(V)χ(F∩ E)dμ_ω(E)


Choose an approximate identity ρ_ϵ on (V), and define ϕ_ϵ= ∫_(V)g^*χ_F ·ρ_ϵ(g)dg∈𝒱^∞(S^2n+1). Then 

    ψ_2k(F)=lim_ϵ→ 0⟨ψ_2k, ϕ_ϵ⟩= lim_ϵ→ 0∫__2n+2-2k(V)ϕ_ϵ(E)  dμ_ω(E)

Now ϕ_ϵ(E)=∫_(V)χ_F(gE)ρ_ϵ(g)dg converges to χ_F∈ C^-∞(_2n+2-2k(V))  in Hörmander's topology on the space of generalized functions with wavefront set contained in (χ_F). It follows that ψ_2k(F)= ∫__2n+2-2k(V)χ_F(E)dμ_ω(E).



 

 §.§ Integral geometry of the contact sphere

 Here we determine the coefficients in Theorem <ref>. We have two different bases of contact-invariant valuations on the contact sphere, indexed by 0≤ k≤ n. Namely, we have ϕ_2k defined in terms of curvature at the contact points, and ψ_2k given by Crofton integrals. Since  ϕ_2k, ψ_2k∈𝒲_2k^-∞(S^2n+1), they are related by a triangular matrix, that is 
    ψ_2k=∑_j=k^n c^n_kjϕ_2j.

 We will compute c_kj^n by evaluating both bases on all great spheres S^2m.
 
 Take F=S^2n=S^2n+1∩Π where Π⊂ S^2n+1 is a fixed hyperplane. It is normally transversal. Note also that χ(F∩ E)=2 for a generic E∈_2n+2-2k(^2n+2), that is χ_F is the constant 2 on _2n+2-2k(^2n+2). Thus by Lemma <ref> we may compute ψ_2k(F) using the explicit Crofton formula. Denote κ_k=min (k, n+1-k). By Lemma <ref>, 
    ψ_2k(S^2n)   =∫__2n+2-2k(^2n+2)χ(S^2n∩ E)σ_ω(E)^-2n-2dσ_1(E)
       =2∫__2n+2-2k(^2n+2)σ_ω(E)^-2n-2dσ_1(E)=(-1)^κ_k2/κ_k!1/(2n+1)^κ_k

 
 Considering spheres of lower dimension, we see that ψ_2k(S^2m)=ψ_2k(S^2n) if m≥ k and zero otherwise. Thus
 
 
    ψ_2k(S^2m)={[ (-1)^κ_k2/κ_k!1/(2n+1)^κ_k,                        k≤ m;                          0,                         k>m ].

 On the other hand by eq. (<ref>), 
 
 
    ϕ_2j(S^2m)=22m  2j

 
 We now plug those values into (<ref>).
 Define the lower-triangular matrix A by A(m, j)=2m 2j, 0≤ m,j≤ n. Its inverse is given by A^-1(j, m)=2j 2mE_2j-2m, where E_i is the i-th Euler (secant) number.
 Set b_k=(-1)^κ_k1/κ_k!1/(2n+1)^κ_k. Then c_kj^n=b_k∑_m=k^jA^-1(j,m). In particular, c^n_kk=b_k, c^n_k, k+1=b_k(1-2k+2 2).
 


 §.§ Contact curvature of convex sets


Here we prove an upper bound on the contact valuations of a convex set.

	Let K⊂  S^2n+1 be a convex subset with C^2 boundary. Then for all 0≤ k≤ n, ϕ_2k(∂ K)≤ϕ_2k(∂ K_0), where K_0 is the hemisphere.  

We will use an auxiliary complex structure: V=^n+1, S^2n+1 is identified with the unit sphere therein.
We may assume that ∂ K is normally transversal. 

First, let us verify that ∂ K has exactly two contact tangent hyperplanes. Since the Euler characteristic χ(∂ K)=2, this amounts to verifying that the intersection index of N^*K and M_H in ℙ_S^2n+1 at every contact point is +1. We will refer to this number as the contact index, denoted I_H(∂ K, x).

Consider a sphere ℰ=S^2n⊂ S^2n+1 given by a quadratic equation in V. It is easy to check by an explicit computation that ℰ has exactly two contact tangent points. 
Since χ(ℰ)=2, we conclude that the contact index of ℰ at each of those points is +1.
Now let x∈𝒦 be a contact tangent point, and let ℰ be the osculating sphere at x. Then I_H(∂ K, x)=I_H(ℰ, x)=1.

Next, let p∈∂ K be a point where T_p∂ K=H_p. We now project the hemisphere U centered at p to ^2n+1=T_pS^2n+1 by a central projection π from the origin π, so that p is mapped to the origin. Clearly π(K∩ U) is convex near the origin: if K=S^2n+1∩ C where C is a convex cone, then π(K∩ U)=C∩ T_pS^2n+1. By a standard computation and assuming p={y_n+1=1}, the resulting contact structure in ^2n+1 is given by the contact form α=-dx_n+1+∑_j=1^n(-y_jdx_j+x_jdy_j). We will write z=-x_n+1.

We thus consider a convex body K with C^2 boundary in the contact space (^2n+1,α). We assume K is tangent to the contact distribution at the origin, which is ^2n={z=0}, and further assume without loss of generality that K lies below it.
The normal to the contact distribution is ν_H=(-y_1,…, -y_n, x_1, …, x_n, 1), the normal to ∂ K is ν_K. Then by Proposition <ref>, 
    ϕ_2n(∂ K, 0)=|(d_0ν_K-d_0ν_H)|^-1D(d_0ν_K-d_0ν_H[2n-2k], J[2k])
 where d_0ν_K, d_0ν_H: ^2n→ T_e_2n+1S^2n, the latter space identified with ^2n. 
Using the coordinates x_1,…, x_n, y_1,…, y_n on ^2n we get 
    d_0ν_H=( [    0 -I_n;  I_n    0 ])=J
  Write also S=d_0ν_K. Thus ϕ_2n(∂ K, 0)=| (S-J)|^-1D(S-J[2n-2k], J[2k])=|(I+SJ)|^-1D(I+SJ[2n-2k], I[2k])=|(I+SJ)|^-1∧^2n-2k(I+SJ). 

Note that S≥ 0 since K is convex. Then JS and √(S)J√(S) have the same characteristic polynomial. The latter matrix is antisymmetric, hence the roots of its characteristic polynomial appear in purely imaginary pairs ± iλ_j, j=1,…, n. Let us write μ_1,…, μ_2n for these eigenvalues in some order. Note that for K_0, all λ_j=0. Writing m=2n-2k, we ought to show that

    ∧^m (I+SJ)≤2n  m(I+SJ)


    ∑_|T|=m∏_t∈ T (1+μ_t) ≤2n  m∏_j=1^n(1+λ_j^2)

We will use induction on n. For n=0,1 the verification is trivial.
Partition the sum over subsets T as S_0+S_1+S_2, where S_j (j=0,1,2) is composed of those summands where ± iλ_1 appears j times inside {μ_t:t∈ T}.
By the induction assumption, S_0≤2n-2 m∏_j=2^n(1+λ_j^2), S_1≤ 2 2n-2 m-1∏_j=2^n(1+λ_j^2), S_2≤2n-2 m-2∏_j=1^n(1+λ_j^2). It remains to check that

    2n-2 m+22n-2 m-1+2n-2 m-2(1+λ_1^2)≤2n m(1+λ_1^2)
 
which clearly follows from the equality

    2n-2 m+22n-2 m-1+2n-2 m-2=2n m

concluding the induction and the proof.



	The case of equality is far from unique: any convex subset which is flat to second order at its two contact points would have the same values of ϕ_2k.





§ SYMPLECTIC INTEGRAL GEOMETRY





 §.§ Symplectic space


Let us first show there is no interesting symplectic valuation theory.

	There is no (2n)-invariant, translation invariant generalized valuation except for linear combinations of χ and _2n.


We will find all invariant k-homogeneous valuations. Let N=⌊min(k,2n-k)/2⌋ be the number of multi-Kähler angles for E∈_k(^2n). Since U(n)⊂(2n), an (2n)-invariant valuations ϕ would also be U(n)-invariant. In particular, it is smooth by Alesker's theorem <cit.>, as U(n) acts transitively on the unit sphere.
We consider two separate cases. When k=2l+1 and E∈_k(V) is maximally non-degenerate, then there is no (E)-invariant Lebesgue measure on E. Since such subspaces are dense in _k(V) we conclude (ϕ)=0, and hence ϕ=0.

When k=2l, an (2n)-invariant section of the Klain bundle should be proportional to |ω^l |_E|, that is after Euclidean trivialization it is proportional to |σ(E)|=∏_i=1^Ncosθ_i. But by <cit.>, the Klain section of a U(n)-invariant valuations should be given by a symmetric polynomial of cos^2θ_i.
Thus there can be no non-trivial (2n)-invariant valuations unless k=0, 2n.


	Instead of using the description of Bernig-Fu, one can simply notice that ∏_i=1^Ncosθ_i∈ C(_k(V)) is not smooth, violating the smoothness of ϕ.

Recall the distributions μ_ω, μ_0∈ℳ^-∞(_2n-2k^+(V))^(V) given by eqs. (<ref>) and (<ref>). 


	For 1≤ k≤ n-1 and a smooth convex body K⊂ V it holds that 
    ∫_^+_2k(V)χ(K∩ E) dμ_0(E)=0.
 

Put differently, μ_0 lies in the kernel of the cosine transform.



Nevertheless, we can write symplectic Crofton formulas with the oriented valuation theory approach detailed in Appendix <ref>.
For a compact oriented C^1-submanifold with boundary F⊂ V of codimension 2k, set _F(E)=I(E, F) for E∈^+_2k(V), which is well-defined whenever E and F intersect transversally.


	For F as above, _F∈ C^-∞(^+_2k(V)), and (_F)∩ N^*X^2k, +_r=∅ for all r<min(k,n-k).




Let Z={(x,E):x∈ E}⊂ V×^+_2k(V) be the incidence manifold, which has a natural orientation, and denote by τ:Z→ V, π:Z→^+_2k(V) the obvious submersions. Consider Z_F=τ^-1(F), which is a C^1 oriented submanifold of Z of codimension 2k. Define δ_Z_F:=τ^*[[F]]=[[Z_F]].  Note that π_*τ^*[[F]]∈ C^-∞(^+_2k(V)), and π_*τ^*[[F]](E)=I(E, F) whenever E⋔ F. The first statement follows. Note that (δ_Z_F)⊂(τ^*), and therefore also π^*(_F)⊂(τ^*).
As π^* is injective, the statement of the lemma would follow from (τ^*)∩π^*N^*X^2k, +_r={0}(τ^*)∩ N^*π^-1X^2k, +_r={0}. 

Take (x,E)∈ Z_F∩π^-1 X^2k, +_r. Define Z̃_x:=τ^-1(x)⊂ Z. Then (τ^*)∩ T^*_x,E=τ^* (T_x^*V)=N^*Z̃_x. Let us check that Z̃_x and π^-1X^2k, +_r intersect transversally at (x, E).
For a tangent vector (v,Ξ)∈ T_x,EZ⊂ T_x V× T_E_2k^+(V), use Lemma <ref> to decompose Ξ=Ξ_x+Ξ_L with Ξ_x∈ T_EZ_x where Z_x:=πZ̃_x, and Ξ_L∈ T_EX^2k, +_r.
In the product manifold V×^+_2k(V) we get the equality (v, Ξ)=(0,Ξ_x)+(v, Ξ_L). 
Since (0, Ξ_x)∈ TZ while by assumption (v, Ξ)∈ TZ, we conclude (v, Ξ_L)∈ TZ. Thus T_x, EZ=T_x, EZ̃_x+T_x,Eπ^-1X^2k, +_r⇒ N^*Z_F∩ N^*π^-1X^2k, +_r={0}, concluding the proof.




 Let F⊂ V be a C^1 compact, oriented submanifold with boundary. Then
	
    ∫_F ω^k=(-1)^κ/2n k2n  2k^-1n^κ∫_^+_2n-2k(V) I(E, F)dμ_ω(E)

	where κ=min(k, n-k).
	

Considered as a function of F, the integral on the right hand side is a Crofton integral as in Proposition <ref>. Hence it defines a closed, 2k-form on V which is (V)-invariant. By the fundamental theorem of invariant theory, it is a multiple of ω^k, that is 

    ∫_^+_2n-2k(V) I(E, F)dμ_ω(E) =C∫_F ω^k


It remains to find the constant C. We will use a compatible Euclidean structure. Let B_W be the unit Euclidean ball in the ω-positively oriented subspace W∈ X^2k_+. We will average the integral over X^2k_+ with respect to the probability measure dW that is invariant under 𝔰𝔬(2n). For easy computation, we replace the exponent of σ_ω in μ_ω(E)=σ_ω(E)|σ_ω(E)|^-2n-1d E with the meromorphic variable s∈ℂ and compute for real s which is sufficiently large so that all integrands are continuous.


    A_s:=∫_  X^2k_+dW∫_^+_2n-2k(V) I(E, B_W)σ_ω(E)|σ_ω(E)|^sdE


    =∫_^+_2n-2k(V)σ_ω(E)|σ_ω(E)|^sdσ_1(E)∫_  X^2k_+dW∫_V/EI(E+x, B_W)dx

Here the inner integral is with respect to dx, the Euclidean Lebesgue measure under the identification V/E=E^⊥. We may write 

    ∫_V/EI(E+x, B_W)dx =σ_ω(E)_2k(Pr_E^⊥(B_W))

Hence 

    A_s= ∫_^+_2n-2k(V)|σ_ω(E)|^sdσ_1(E)∫_  X^2k_+_2k(Pr_E^⊥(B_W))dW

The inner integral is independent of E and can be computed using the Kubota formula. 

    ∫ __2n-2k(V)_2k(Pr_F^⊥ (B^2k))dσ_1(F) =c_0 _2k(B^2k)=c_0 π^k/k!

where B^2k is any fixed 2k-dimensional Euclidean ball, and 
    c_0=2n  2k=2n  2k_2n(B^2n)/_2k(B^2k)_2n-2k(B^2n-2k)=2n  2kn k^-1.
  We get

    A_s=c_0 π^k/k!  2∫_[0,1]^κ (λ_1…λ_κ)^sdλ_1… dλ_κ = 2c_0 π^k/k!1/(s+1)^κ

and taking s=-(2n+1) we conclude 
A=2c_0(-1)^κπ^k/k!1/(2n)^κ.
Averaging the right hand side of eq. (<ref>), we get

    C ∫_X_2k^+(B^2k)|σ_ω(W)|dW
      = Cπ^k/k!∫_[0,1]^κλ_1…λ_κ dλ_1… dλ_κ =Cπ^k/k!2^κ.

Summing up, 2c_0 π^k/k! (-1)^κ1/(2n)^κ =π^k/k!2^κC so that

    C^-1=(-1)^κ(2n)^κ/2^κ+1 c_0=(-1)^κ/2n k2n  2k^-1n^κ






	
	

§ ORIENTED VALUATION THEORY

	In this appendix we draw a common thread between valuation theory and the much simpler theory of closed differential forms and linking integrals.
	
		Let X^n be an oriented manifold. We will think of the closed k-forms, denoted 𝒵_k(X), as smooth oriented valuations of degree k, and consider them as functions on k-dimensional oriented submanifolds of X with boundary, given by integration: ω(A)=∫_Aω. The form is clearly determined by its value on all submanifolds,  which is analogous to the Klain embedding.
		Moreover, ω(A) only depends on ∂ A.
		
		The wedge product on 𝒵(X) turns it into an algebra. When X is compact, we have Poincaré duality, namely that 𝒵_k(X)⊗𝒵_n-k(X)→𝒵_n(X)=Ω^n(X)→ is non-degenerate, where the last arrow is given by ∫_X.
		
		We also consider the closed currents, which are analogous to the generalized valuations. We will denote them by 𝒵^-∞(X). We will sometimes write 𝒵^∞ instead of 𝒵.
		
		Now assume X=V=^n. The translation-invariant (smooth or generalized) oriented valuations 𝒵^±∞(V)^tr are just ∧^∙ V^*. The following construction appears in <cit.>.
		
		For oriented manifolds with boundary A,B⊂ V of complementary dimensions, at least one of which is compact, let I(A, B) denote their intersection index. It is well-defined when A, B are in general position. 
		Note that for a closed (as a subset) (n-k)-dimensional submanifold E⊂ V, I(∙, E)∈𝒵^-∞_k(V).
		
		Given a distribution μ∈ℳ^-∞(^+_n-k(V)), we define (μ)∈𝒵^-∞_k(V) by (μ)=∫_^+_n-k(V) I(∙, E)dμ(E). Clearly the even measures (with respect to the orientation-reversing map) lie in the kernel of , so we restrict our attention to odd measures.
		
		
			The map : ℳ^±∞(^+_n-k(V))^tr→𝒵_k(V)^tr is surjective.
		
		The (V)-module Z_k(V)^tr=∧ ^k V^* is irreducible. Since the Crofton map is (V)-equivariant, it suffices to show  is non-zero. This is not hard to see, for instance (δ_E-δ_-E)(A)=2I(A, E), where ⟨δ_E, f⟩=∫_V/Ef(x+E)dx on a compactly supported test function f. For a smooth example, one could convolve with an approximate identity on (V).
		The analogues of the the Alesker-Poincaré and Alesker-Fourier dualities coincide in this setting:
		The Alesker-Poincaré pairing is the wedge product ∧^kV^*⊗∧^n-kV^*→∧^nV^*.
		The Alesker-Fourier duality operation is given by the Hodge star: ∗: ∧^kV^*→ (∧^n-kV^*)^*⊗∧^n V^*. 
		
		The following easy statement is the analogue for oriented valuations of the principal kinematic formula. Due to the finite-dimensionality of the space of translation-invariant forms, one can average over translations alone.
		
		Let a top form _n∈∧ ^n V^* be fixed on V. Let Ω∈∧^∙ V^*⊗∧^∙ V^*=(∧^∙ V⊗∧^∙ V)^* be the wedge product, that is for ξ, η∈∧ ^∙ V, ⟨ξ∧η, _n⟩ =Ω(ξ,η).
		
		For oriented compact manifolds with boundary A, B we write Ω(A,B):=∫_A× BΩ, which can be written more explicitly by representing Ω=∑ω_i⊗ω'_i, then Ω(A, B)=∑∫ _A ω_i∫_Bω'_i (the integrals with mismatched dimension vanish by definition). Consider the kinematic operator
		
    K_V(A,B):=∫_V I(A, B+x)d_n(x)

		which is well-defined, since the integrand is compactly supported.
		
			Let A^n-k, B^k⊂ V be compact, oriented submanifolds with boundary, of complementary dimension. Then K_V(A, B)=Ω(A, B)
		
		Let [[B]] be the current defined by B. Then [B]_V:=∫_V [[B+x]]d_n(x) is translation-invariant, that is [B]_V∈∧^kV. Now I(A, B)=[[A]]∩ [[B]], so that K_V(A, B)=[[A]]∩[B]_V=_n([A]_V∧[B]_V)=Ω(A, B) as claimed.
		
		
		We remark that this formula is also reminiscent of the Bezout formula in complex algebraic geometry.
		
	
	

§ INVARIANT SECTIONS

	We will need two technical lemmas concerning invariants of group actions.
	
	The first goes back to Kolk and Varadarajan <cit.>, and appeared in a form best suitable for our needs in <cit.>. Let us quote the result in its simplest sufficient form.
	
	
	Take a Lie group G acting on a manifold X with finitely many orbits, all locally closed submanifolds. Let E be a G-vector bundle over X. Define for integer α≥ 0 and a submanifold Y⊂ X the G-bundle F^α_Y over Y by 
	F^α_Y|_y=E|_y ⊗^*(N_yY) ⊗^α(N_yY) .                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
	A generalized section s∈Γ^-∞_Y(X,E) has a certain transversal order α along Y, and a transversal principal symbol σ(s)∈Γ^-∞(Y, F^α_Y). For details, see e.g. <cit.>.
		
		Let Z⊂ X be a closed G-invariant subset.
		Decompose into G-orbits: Z=⋃_j=1^J Y_j. Then
		
		Γ^-∞_Z(X,E)^G ≤∑_α=0^∞∑_j=1^J Γ^∞(F^α_Y_j)^G.
		
	
	

	The second statement is surely well-known, but we were not able to locate a reference. For completeness, we include the proof.
	
	Let G be a (possibly infinite dimensional) Lie group acting on a manifold M^n transitively, and let ℰ be an infinite-dimensional G-bundle of Fréchet spaces over X. Then the space of G-invariants of Γ_c^∞(M, ℰ)^* is naturally isomorphic to Γ^∞(M, ℰ^*⊗ |ω_M|)^G

 In the infinite dimensional case, we assume that every T_x M admits a basis of infinitesimal generators of 1-parametric subgroups of G. This is certainly the case for Diff(M), Symp(M), Cont(M).

First we consider the wavefront set of the average of a distribution along a flow.	
	
		Let M be a manifold. Assume  acts on M, and let the curve C⊂ M be an orbit.
		Let ϕ∈ C^-∞(M) be a generalized function, and μ∈ C^∞_c(). 
		Define ϕ∗μ:=∫_ t^*ϕ·μ(t)dt. Then for all p∈ C we have (ϕ∗μ)∩ T^*_pM⊂ N_p^*C.
	
Fix a small neighborhood V of p∈ C which is -equivariantly identified with a neighborhood U⊂^n of p=0, with  acting by translations along the x_1-axis and C coinciding with the x_1-axis.
Choose a partition of unity ρ_i on M and w_j on . Writing ϕ_i:=ρ_i·ϕ, μ_j:=w_j·μ we have ϕ∗μ=∑_i,jϕ_i∗μ_j.  We may assume that each ϕ_i has finite order, and that each ψ=ϕ_i∗μ_j whose support contains p, is in fact supported inside V. Taking such ψ, we may write
ψ=Φ∗π_1^*ν for some ν∈ C^∞_c() and Φ∈ C_c^-∞(^n) of order k, where π_1:^n→ is the projection to the first coordinate.
Then the Fourier transform satisfies ψ(ξ) =Φ(ξ)ν(ξ_1).
Now |Φ(ξ)|≤ C(1+|ξ|)^k while |ν(ξ_1)|≤ C_N(1+|ξ_1|)^-N for all N. It follows that for all ϵ>0, the cone
C_ϵ:={ξ_1^2>ϵ(ξ_2^2+…+ξ_n^2)} falls outside the wavefront set of ψ, so that (ψ)⊂{ξ_1=0}.
That concludes the proof.
Clearly the same statement holds also for S^1-actions, and with ϕ a generalized section of an arbitrary -equivariant vector bundle over M.

	Proof of Proposition <ref>.
	Take s∈Γ_c^∞(M, ℰ)^∗. For ϕ∈Γ_c^∞(M,ℰ) one can define s·ϕ∈ℳ^-∞(M) by ∫ f· d(s·ϕ):=s(fϕ) for f∈ C^∞_c(M).
	Moreover, the map Γ_c^∞(M, ℰ)→ℳ^-∞(M) given by ϕ↦ s·ϕ is G-equivariant.
	Fix ϕ, and let us verify that s·ϕ is in fact a smooth measure. 
	Consider a 1-parametric subgroup H⊂ G which can be either  or S^1. By the Dixmier-Mallavin theorem, we may find smooth probability measures μ_1,…,μ_N∈ℳ^∞_c(H) and sections ψ_1,…,ψ_N∈Γ_c^∞(M,ℰ) such that 
    ϕ=∑_j ∫_H gψ_j dμ_j(g)⇒ s·ϕ=∑_j ∫_H g(s·ψ_j)dμ_j(g)
 
 	It follows by Lemma <ref> that the wavefront set (s·ϕ) lies in the conormal bundle to the orbits of H on M. Since G acts transitively, we conclude (s·ϕ)=∅, that is s·ϕ∈ℳ^∞(M).
	
	In particular, we can consider the density on every tangent plane, (s·ϕ) (x)∈(T_xM).
	We next claim that (s·ϕ)(x) only depends on ϕ(x). Indeed, if ϕ(x)=0 we may represent ϕ=f·ψ for some ψ∈Γ_c^∞(M,ℰ) and f_c∈ C^∞(M) with f(0)=0. Then (s·ϕ)=f· (s·ψ), and thus (s·ϕ)(x)=0. That is, we get an element s(x)∈ℰ^*|_x⊗(T_xM). By G-invariance, x↦ s(x) is a smooth section of ℰ^*⊗ |ω_M|, which clearly defines the same functional as s on Γ_c^∞(M, ℰ). 


amsalpha

