Geometric measures of uniaxial solids of revolution in ${\mathbb{R}^{4}}$ and their relation to the second virial coefficient
Markus Kulossa Joachim Wagner




Geometric measures of convex uniaxial solids of revolution in ℝ^4]Geometric measures of uniaxial solids of revolution in ℝ^4 and their relation to the second virial coefficient



markus.kulossa@uni-rostock.de

Joachim Wagnerjoachim.wagner@uni-rostock.de


Institut für Chemie, Universität Rostock, 18051 Rostock, Germany


We provide analytical expressions for the second virial coefficients of hard, convex, monoaxial solids of revolution in ℝ^4. 
The excluded volume per particle and thus the second virial coefficient is calculated using quermassintegrals and rotationally invariant mixed volumes based on the Brunn-Minkowski theorem. 
We derive analytical expressions for the mutual excluded volume of four-dimensional hard solids of revolution in dependence on their aspect ratio ν including the limits of infinitely thin oblate and infinitely long prolate geometries.  Using reduced second virial coefficients B_2^*=B_2/V_ as size-independent quantities with V_ denoting the D-dimensional particle volume, the influence of the 
particle geometry to the mutual excluded volume is analyzed for various shapes. Beyond the aspect ratio ν, the detailed particle shape influences the reduced second virial coefficients B_2^*.
We prove that for D-dimensional spherocylinders in arbitrary-dimensional Euclidean spaces ℝ^D their excluded volume solely depends on at most three intrinsic volumes, whereas for different convex geometries
D intrinsic volumes are required. For D-dimensional ellipsoids of revolution, the general parity B_2^*(ν)=B_2^*(ν^-1) is proven.







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    April 29, 2024
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§ INTRODUCTION



Place two identical coins on a table and roll one coin completely around the second coin's rim: The moving coin completes two revolutions during the turn around the
first coin which is known as the coin rotation paradox. 
To analyze this phenomenon, in addition to a cardioid curve generated by a point on the perimeter of the second coin, the excluded volume between two rigid bodies can be used: 
The excluded volume is the volume inaccessible for the second body in the vicinity of the first one. The perimeter of the excluded volume is the curve generated
by the center of the second body during its turn around the first one preserving contact distance. 
As visualized in Fig. <ref>, for two circles with radius r_0 the two-dimensional excluded volume (or area) is a circle with radius 2r_0 and area 4π r_0^2, i.e.,
per circle 2π r_0^2  and thus twice the two-dimensional volume of one circle. Hence, the number of revolutions of the turning coin is the ratio of the excluded volume's perimeter
to the coin's perimeter.

While this problem can be generalized to coins with unequal radii, also rolling a disk around other, possibly anisotropic shapes is feasible. 
When both objects are anisotropic, the excluded area additionally depends on the relative orientation of these bodies as shown in Fig. <ref> (right) for stadia. 
Here, an orientation-averaged, two-dimensional excluded volume results which can be generalized to arbitrary, convex hard bodies in Euclidean spaces ℝ^D.

The excluded volume of hard solids has implications on the physics of multi-particle systems consisting of such objects: The phase behavior and equation of state of
hard-body systems are governed by their geometry <cit.>. 
Kamerling-Onnes <cit.> proposed the virial expansion


    Z    =pϱ T = 1 + B_2 ϱ + B_3 ϱ^2 + … = 1 + ∑_i=2^∞ B_i ϱ^i-1


for the real gas factor Z of imperfect gases as a heuristic equation of state, an expansion in powers of the number density ϱ 
where p denotes the pressure,  Boltzmann's constant, and T the temperature. The coefficients B_i are
virial coefficients of order i. Introducing the packing fraction η=ϱ V_ P with V_ P denoting the D-dimensional
particle volume, Eq. (<ref>) can be reformulated as an expansion in powers of the packing fraction η


    Z    = p V_η T = 1 + B_2^*η + B_3^*η^2 + … = 1 + ∑_i=2^∞ B_i^* η^i-1


with the dimensionless, reduced virial coefficients B_i^*=B_i/V_ P^i-1 as expansion coefficients. Introducing a generalized pressure as
ratio of energy and D-dimensional volume, the virial expansion can be used in Euclidean spaces ℝ^D with arbitrary dimension D.


 




The first virial coefficients were calculated for hard spheres in ℝ^3. Starting point is the calculation of the second virial coefficient by Jäger <cit.>. Later, Boltzmann, van der Waals, and van Laar provided analytical solutions for virial coefficients of hard spheres up to the fourth order <cit.>. Up to now, analytical solutions for virial coefficients of order five and higher are not available. Later on, analytical expressions for virial coefficients of isotropic particles in ℝ^2 <cit.> and Euclidean spaces with dimension D>3 <cit.> were reported up to the fourth order.  

Starting point for studies of non-spherical particles is the seminal work of Onsager <cit.> with infinitely thin hard rods as a model system for liquid crystals <cit.>.
Few years later, Isihara and Hadwiger independently provided an analytical expression for the second virial coefficient of arbitrary convex particles in ℝ^3 <cit.>. The analogue 
for convex solids in ℝ^2 is provided by Boublík <cit.>. For concave geometries in ℝ^2,ℝ^3, and ℝ^4 only few analytical or semianalytical expressions
for the second virial coefficient are available <cit.>. For selected concave, rigid shapes, numerical values for their second virial coefficients as the mutually excluded volume per particle have been provided <cit.>. 

Higher order virial coefficients, even for convex, anisometric particles are only numerically available, e.g., by means of Monte Carlo integration <cit.>. 
Analytical expressions for the second virial coefficients of solids of revolution in ℝ^3 employing the Isihara-Hadwiger theorem are provided in a previous work <cit.>.
Recently, the Brunn-Minkowski theorem has been applied to calculate second virial coefficients of various convex particles in Euclidean spaces with dimension D>3 <cit.>. 

In this work, we use the Brunn-Minkowski theorem to provide analytical expressions for the second virial coefficients B_2 of convex, uniaxial solids of revolution in ℝ^4. 
Therefore, analytical expressions for their quermassintegrals (and intrinsic volumes) are derived depending on their meridian curve. The results for selected geometries are compared and the detailed influence of the particle shape is analyzed. The comparison of solids of revolution with identical meridian curve in Euclidean spaces with different dimensionality can provide fruitful insights to the self organization of condensed matter in ℝ^3. Beyond the impact on real physical systems, e.g., equation-of-state data and phase behavior, insights into close packings in higher dimensional spaces can be expected based on the provided geometric measures <cit.>. 





§ THEORETICAL BACKGROUND
 



 §.§ Virial theory

Mayer and Mayer have shown by means of statistical mechanics, starting from the grand canonical partition function, that the virial coefficient of order i depends on interactions in an i-particle
cluster <cit.>. The second virial coefficient, describing the initial departure from the ideal-gas behavior in the low-density limit, reads as


    B_2 = -12V(Z_2 - Z_1^2 )


with Z_N denoting the configuration integral for N particles in a system volume V. Let U(r⃗_1,r⃗_2) be the potential energy of two particles with centers located at r⃗_1 and r⃗_2, the second virial coefficient can be written as 


    B_2 = -12V∬{exp[-U(r⃗_1,r⃗_2)]-1 } ^D r⃗_1  ^D r⃗_2 .


With the coordinate system's origin at r⃗_1 and the distance vector r⃗_12=r⃗_2-r⃗_1,
performing the integration over ^Dr⃗_1, the second virial coefficient can be reformulated as


    B_2 = - 12∫{exp[-U(r⃗_12)]-1 } ^D r⃗_12 .


Here, the integration over ^D r⃗_1 with


    ^D r⃗   = r^D-1  r ∏_i=0^D-2sin^i ( ϕ_i )  ϕ_i = ^D V

as D-dimensional volume element using polar coordinates results in the system volume V.
The integrand in Eq.(<ref>) 

    f_12 = exp[-U (r⃗_12)]-1

is the Mayer f function depending on the interaction of both particles. For hard-body interaction, the
potential reads as 


    U(r⃗_12) = {[        ∞        : r_12 < σ;        0        :   r_12≥σ;          ].
 

with σ denoting the contact distance which in the case of uniaxial, anisometric shapes depends on the unit vectors 
û⃗̂_1, û⃗̂_2 denoting their orientation, and the direction of the distance vector r̂⃗̂_12=r⃗_12/r_12.
Hence, combining Eqs. (<ref>) and (<ref>) results in 


    f_12 = {[       -1        : r_12 < σ;        0        :   r_12≥σ;          ].
 

for the Mayer f function, irrespective of the thermal energy T: For an overlap, f_12=-1 is obtained while this integrand vanishes for non-overlap configurations.

For D-dimensional spheres with radius r_0 as radially symmetric bodies, the contact distance is, independent of the orientation, σ=2 r_0,
leading to 


    B_2 = 12β_D ∫_0^σ r^D-1 r


with 


    β_D = 2 π^D/2Γ(D/2)


denoting the surface area of a D-dimensional sphere with radius r_0=1 where


    Γ(z) = ∫_0^∞ t^z-1exp( -t )   t


denotes the Euler gamma function. Performing the remaining integration over r leads
to


    B_2 = 12κ_D (2 r_0)^D


with the volume 


    κ_D = π^D/2Γ (1+D/2)


of the D-dimensional unit sphere for the second virial coefficient of a D-dimensional sphere.

Since in the case of uniaxial, anisometric shapes the contact distance depends on the orientations û⃗̂_1, û⃗̂_2, and r̂⃗̂_12, the 
orientation-averaged Mayer f function has to be used as integrand leading to


    B_2 = - 12∫⟨ f_12⟩_û⃗̂_2 ^D r⃗_12


where the direction û⃗̂_1 defines the orientation of the coordinate system. 

For hard-body interaction, the second virial coefficient 


    B_2 = 12 V_


is the mutual excluded volume per particle and thus solely related to geometric properties of the considered shape.






 §.§ Excluded volumes of convex solids


Let K,L∈ℝ^D be two convex bodies. The rotation-averaged excluded volume V_(K,L) is


    V_ (K,L)=1κ_D∑_i=0^D Di  W_i(K)  W_D-i(L) ,


known as Brunn-Minkowski theorem <cit.>.
The quantity W_i(K) is the mixed volume of (D-i) convex bodies K and i D-dimensional unit spheres 
denoted as i-th quermassintegral of K which relates to the intrinsic volume υ_i as

    υ_i (K) = DiW_D-i(K)κ_D-i

for a convex set. 

If L is a D-dimensional sphere with radius ε, using 


    W_i^( sph)(ε)   = κ_D ε^D-i


as a sphere's quermassintegral of order i <cit.>, Eq. (<ref>)
reads as


    V_(K,ε)    = ∑_i=0^D Di W_i(K)ε^i ,


also known as Steiner formula <cit.>. This excluded volume V_(K,ε) is also the volume of K's ε-neighborhood as visualized in Fig. <ref>. 



For identical convex bodies K=L, the Brunn-Minkowski theorem reads as 


    V_(K) = V_(K,K) = 1κ_D∑_i=0^D Di W_i(K) W_D-i(K)


and thus provides an analytical expression for the second virial coefficient B_2(K) =V_(K)/2 of a convex particle since due to the rotational invariance of quermassintegrals directly the
orientation average is obtained.

Quermassintegrals W_i of orders i={0,1} and i={D-1,D} are related to common geometric measures as



    W_0(K)    = V_(K) , 
    
    W_1(K)    = 1D S_(K) , 
    
    W_D-1(K)    = R̃_(K) κ_D , 
    
    W_D(K)    = κ_D



where V_ denotes the volume of K, S_ its total surface area, and R̃_ its mean radius of curvature <cit.>.
 

Combining Eqs. (<ref>), (<ref>), (<ref>),  and (<ref>), in the two- and three-dimensional Euclidean space, virial coefficients for arbitrary convex solids K
can be calculated. In the two-dimensional Euclidean space ℝ^2, 


    B_2(K) = V_(K) + 12 S_(K) R̃_(K) = V_(K) + S_(K)^24 π


is obtained <cit.>. The relation for the three-dimensional Euclidean space ℝ^3


    B_2(K)    = V_(K) + S_(K) R̃_(K)


is known as Isihara-Hadwiger theorem, independently derived by Isihara <cit.>  and Hadwiger <cit.>.  
Using these relations, second virial coefficients for various convex shapes in ℝ^2 and ℝ^3 have been calculated <cit.>.

In ℝ^4, the second virial coefficient reads as


    B_2(K)    = V_(K) + S_(K)  R̃_(K) + 6π^2 W_2^2(K) .


To calculate W_2(K), the surface integral over continuous surface curvature


    W_2(K)    = 1D∮ H_(K)  ^D-1 S


is required where H_ is the mean curvature


    H_   = 1D-1∑_i=1^D-11R_i


depending on the principal radii of curvature R_i of a convex solid <cit.>.






§ GEOMETRIC MEASURES OF FOUR-DIMENSIONAL UNIAXIAL SOLIDS OF REVOLUTION


Let K be a convex particle in ℝ^4. Its second virial coefficient B_2(K) can analytically be calculated with Eq. (<ref>) using its geometric measures volume V_(K), surface area S_(K), second quermassintegral W_2(K), and mean radius of curvature R̃_(K). 
Uniaxial solids of revolution with meridian curve r(z) can be parameterized conveniently in hypercylindrical coordinates  


    Ψ = [ w; x; y; z;   ]= [ r sinϑsinχcosφ; r sinϑsinχsinφ;     r sinϑcosχ;         r cosϑ;                ] = [ r(z) sinχcosφ; r(z) sinχsinφ;     r(z) cosχ;             z;               ] ,


related to Cartesian or polar coordinates.
Since the volume of a uniaxial solid of revolution in ℝ^D is simply


    V_(K) = κ_D-1∫ r^D-1(z)  z ,


using hypercylindrical coordinates, the expression


    V_(K) 
    
    	     = 43π∫_z_ min^z_ max r^3(z)  z


results in ℝ^4 for a particle with length  l=z_ max-z_ min.
If r(z_ min)=r(z_ max)=0, its surface area can be written as


    S_(K) = β_D-1∫ r^D-2(z) {1+[ r(z) z]^2 }^1/2  z


in ℝ^D with 


    s    = {[ r(z)]^2 + [ z]^2 }^1/2 
    = {1+[ r(z) z]^2 }^1/2  z


being the infinitesimal arc length element of the meridian curve with the result


    S_(K)    = 4 π∫_z_ min^z_ max r^2(z) [1+ṙ^2(z) ]^1/2  z


in ℝ^4 <cit.>. 


The second quermassintegral W_2(K) depends on the principal radii of curvature R_i of the convex solid [Eqs. (<ref>) and (<ref>)],
accessible as reciprocal eigenvalues of the Weingarten map 


    𝒲   = I_^-1II_ .


The Weingarten map 𝒲 is also known as shape operator where I_ and II_ denote the first and second fundamental form of the solid's surface Ψ.


Using the first derivatives


    ∂Ψ∂φ   = [ -r(z) sinχsinφ;  r(z) sinχcosφ;              0;              0 ] ,  ∂Ψ∂χ = [ r(z) cosχcosφ; r(z) cosχsinφ;    -r(z) sinχ;             0 ] ,
    
     ∂Ψ∂ z = [ ṙ(z) sinχcosφ; ṙ(z) sinχsinφ;     ṙ(z) cosχ;             1 ]


the first fundamental form 


    I_ = [   (∂Ψ∂φ) ·(∂Ψ∂φ)   (∂Ψ∂φ) ·(∂Ψ∂χ)  (∂Ψ∂φ) ·(∂Ψ∂ z);   (∂Ψ∂χ) ·(∂Ψ∂φ)   (∂Ψ∂χ) ·(∂Ψ∂χ)  (∂Ψ∂χ) ·(∂Ψ∂ z);  (∂Ψ∂ z) ·(∂Ψ∂φ)  (∂Ψ∂ z) ·(∂Ψ∂χ) (∂Ψ∂ z) ·(∂Ψ∂ z);                  ]


of a uniaxial solid of revolution in ℝ^4 reads as 


    I_ = [ r^2(z) sin^2 χ              0              0;              0         r^2(z)              0;              0              0       1+ṙ^2(z);                ] .

The second fundamental form can be written as 


    II_ = [   n̂⃗̂·∂^2 Ψ∂φ^2   n̂⃗̂·∂^2 Ψ∂φ∂χ  n̂⃗̂·∂^2 Ψ∂φ∂ z;   n̂⃗̂·∂^2 Ψ∂χ∂φ   n̂⃗̂·∂^2 Ψ∂χ^2  n̂⃗̂·∂^2 Ψ∂χ∂ z; n̂⃗̂·∂^2 Ψ∂ z ∂φ n̂⃗̂·∂^2 Ψ∂ z ∂χ  n̂⃗̂·∂^2 Ψ∂ z^2;                  ]


with the second derivatives 



    ∂^2 Ψ∂φ^2   = [ -r(z) sinχcosφ; -r(z) sinχsinφ;              0;              0 ] ,    ∂^2 Ψ∂φ∂χ = ∂^2 Ψ∂χ∂φ = [ -r(z) cosχsinφ;  r(z) cosχcosφ;              0;              0 ] , 
    
    ∂^2 Ψ∂χ^2   = [ -r(z) sinχcosφ; -r(z) sinχsinφ;     -r(z) cosχ;              0 ] ,    ∂^2 Ψ∂χ∂ z  = ∂^2 Ψ∂ z ∂χ = [ ṙ(z) cosχcosφ; ṙ(z) cosχsinφ;    -ṙ(z) sinχ;             0 ] , 
    
    ∂^2 Ψ∂ z^2   = [ r̈(z) sinχcosφ; r̈(z) sinχsinφ;     r̈(z) cosχ;              0 ] ,     ∂^2 Ψ∂φ∂ z  = ∂^2 Ψ∂ z ∂φ = [ -ṙ(z) sinχsinφ;  ṙ(z) sinχcosφ;              0;              0 ] ,



and the normal field n̂⃗̂≡n̂⃗̂(φ, χ, z) of the solid's surface. 
 
A normal, perpendicular to three derivatives a⃗=∂Ψ/∂φ, b⃗=∂Ψ/∂χ and c⃗=∂Ψ/∂ z can in ℝ^4 be written as



    n⃗(a⃗,b⃗,c⃗) = ([ e_w a_w b_w c_w; e_x a_x b_x c_x; e_y a_y b_y c_y; e_z a_z b_z c_z;     ])


using the canonical basis (e_w,e_x,e_y,e_z) with the parities 


    n⃗(a⃗, b⃗, c⃗)    = - n⃗(a⃗, c⃗, b⃗) 
    
    = n⃗(b⃗, c⃗, a⃗)   = - n⃗(b⃗, a⃗, c⃗) 
    
    = n⃗(c⃗, a⃗, b⃗)    = - n⃗(c⃗, b⃗, a⃗).


A uniaxial solid's of revolution second fundamental form using the normal field 


    n̂⃗̂ = n⃗‖n⃗‖
    
     = 1[1+ṙ^2(z) ]^1/2[ -sinχcosφ; -sinχsinφ;     -cosχ;      ṙ(z);           ]


reads as


    II_ = 1[1+ṙ^2(z) ]^1/2[ r(z) sin^2 χ            0            0;            0         r(z)            0;            0            0       -r̈(z);              ] .


From both fundamental forms, the Weingarten map 


    𝒲 = [   1r(z) [1+ṙ^2(z)]^1/2                      0                      0;                      0  1r(z) [1+ṙ^2(z) ]^1/2                      0;                      0                      0 - r̈(z)[1+ṙ^2(z) ]^3/2;                        ]


is obtained as a main diagonal matrix. Hence, the eigenvalues are directly accessible from the diagonal elements indicating the principal curvatures.
The mean curvature thus can be written with the trace of the Weingarten map as 


    H_ = 1D-1(𝒲) =  1D-1(I_^-1II_) .


Hence, for a uniaxial solid of revolution in ℝ^4 three principal radii of curvature 


    R_1(z)    = R_2(z) = r(z) [1+ṙ^2(z)]^1/2
    
    R_3(z)    = - 1r̈(z)[1+ṙ^2(z)]^3/2


result as inverses of the principal curvatures where two of them are identical.

Combining Eqs. (<ref>), (<ref>), and (<ref>), the mean curvature can be written as 


    H_ = 13[1R_1(z) + 1R_2(z) + 1R_3(z)]


in dependence on the meridian curve r(z). Using the surface element ^3 S = r^2(z) sinχ φ χ  s, the second quermassintegral in ℝ^4 with Eqs. (<ref>) and (<ref>), performing the integration over the angular coordinates φ and χ, finally reads as 


    W_2(K) = π3∫_z_ min^z_ max  r^2(z)  [1R_1(z)+1R_2(z)+1R_3(z)] [1+ṙ^2(z) ]^1/2  z


for convex solids with continuous surface curvatures.

The mean radius of curvature of a convex solid with continuous surface curvature can, in general, be calculated from its principal radii of curvature as 


    R̃_ = 1β_D∮1D-1[ ∑_i=1^D-1 R_i(ϕ_0, ϕ_1, …ϕ_D-2) ] ∏_j=0^D-2sin^j (ϕ_j ) ϕ_j


normalized to the surface area of a D-dimensional unit sphere.
In ℝ^4,


    R̃_(K) = 16 π^2∫_0^π∫_0^π∫_0^2π[R_1(ϑ, χ, φ) + R_2(ϑ, χ, φ) + R_3(ϑ, χ, φ) ] sin^2 ϑsinχ φ χ ϑ


is obtained. Employing the surface element 


    ^3 S = R_1 R_2 R_3 sin^2 ϑsinχ φ χ ϑ = r^2(z) sinχ[1+ ṙ^2(z) ]^1/2 φ χ  z ,


the angular integral can be written as a surface integral 


    R̃_(K) = 23π∫_z_ min^z_ max r^2(z) [1R_2(z) R_3(z)+1R_1(z) R_3(z)+1R_1(z) R_2(z)]  [1+ṙ^2(z) ]^1/2  z


where again the integration over the angular coordinates φ and χ results in a factor 4π.


Using the relations to the other quermassintegrals in Eq. (<ref>), analytical expressions for them can be summarized as 



    W_0(K)    = V_(K) 
    
    W_1(K)    = 14∮ ^3 S = 14 S_ (K) 
    
    W_2(K)    = 14∮13[1R_1 + 1R_2 + 1R_3] ^3 S 
    
    W_3(K)    = 14∮13[1R_1 R_2 + 1R_1 R_3 + 1R_2 R_3] ^3 S = 14R̃_β_4 = R̃_κ_4 
    
    W_4(K)    = 14∮1R_1 R_2 R_3 ^3 S = 14β_4 = κ_4



for convex solids with continuous surface curvature in ℝ^4. 

For arbitrary Euclidean spaces ℝ^D, this can, in general, be written as 


    W_i(K) = {[                                              V_(K)                                                  :                                                i=0;                                          1D ^D-1 S                                                  :                                                i=1; 1D[[ D-1; i-1 ]]^-1[∑_j=1^D-i+1 a_D-i+1,j]  ^D-1 S                                                  :                                          2 ≤ i ≤ D ].


with 


    a_ij = {[                    1R_j                       :                   i=D-1;     1R_j∑_k=j+1^D-11R_k                       :                   i=D-2; 1R_j∑_k=j+1^i+1 a_i+1,k                       :                   i<D-2 ].


and 


    ^D S = [ ∏_i=1^D R_i ] [ ∏_j=0^D-1sin^j(ϕ_j)  ϕ_j ]


for the surface element. 

For convex solids with discontinuities of their surface curvature, additional contributions to quermassintegrals W_i(K) need to be considered:
Additional contributions to W_1(K) exist for uniaxial solids of revolution in ℝ^4 if r(z_ min)≠ 0 or r(z_ max)≠ 0. One- and two-dimensional singularities of the surface curvature where in the case of uniaxial solids of revolution in ℝ^4 only two-dimensional singularities can exist 
cause additional contributions to W_2(K) and W_3(K). Zero-dimensional singularities which can for uniaxial solids of revolution only exist at r(z)=0 do not cause additional contributions to W_i for 0≤ i≤ D.   
In the following, we provide quermassintegrals for various uniaxial solids of revolution in ℝ^4 including shapes with discontinuities in their surface curvature.







§ SPECIFIC GEOMETRIC MEASURES FOR SELECTED SOLIDS OF REVOLUTION


In the following subsections, we provide the geometric measures volume V_, total surface area S_, second quermassintegral W_2,
and mean radius of curvature R̃_ for selected convex geometries. 
Uniaxial solids of revolution in ℝ^4 are defined by their meridian curve r(z).
Based on the meridian curve r(z), specific geometric measures of these geometries and the related, 
second virial coefficients are analytically accessible employing Eqs. (<ref>), (<ref>), and (<ref>).




 §.§ Solids of revolution with continuous surface curvature




  §.§.§ Hypersphere


The simplest solid of revolution in the four-dimensional Euclidean space ℝ^4 is the isotropic hypersphere. 
Placing the center of a hypersphere in the coordinate system's origin, the meridian curve r(z) of the hypersphere reads as 


    r(z) = (r_0^2 -z^2 )^1/2


with r_0 being the radius of the hypersphere and z the coordinate on the axis of revolution resulting from hypercylindrical coordinates [Eq. (<ref>)]. 
Using Eq. (<ref>), the hypervolume of a hypersphere can simply be written as 


    V_   = 43π∫_-r_0^r_0(r_0^2-z^2 )^3/2  z =  12π^2 r_0^4


which is identical to V_ = κ_4 r_0^4 using Eq. (<ref>). 
The total area S_ of a surface without singularities is simply the lateral surface area S_^'. 
The lateral surface area S_^' of a hypersphere can be calculated with Eqs. (<ref>) and  (<ref>) 
leading to 


    S_   = S_^' = 4 π r_0  ∫_-r_0^r_0(r_0^2-z^2 )^1/2  z =2 π^2 r_0^3


as the expected results, identical to S_ = β_4 r_0^3 using Eq. (<ref>) with D=4. 

For the second quermassintegral W_2 and the mean radius of curvature R̃_ of a four-dimensional, convex solids of revolution the principal radii of curvature R_1, R_2, and R_3 are required [Eqs. (<ref>) and (<ref>)]. 
These can be calculated in dependence on the meridian curve as 


    R_1 = R_2 = R_3 = r_0


for a hypersphere employing Eqs. (<ref>) and (<ref>).  
Its principal radii of curvature are simply the radius of the hypersphere r_0.  
For the second quermassintegral W_2, Eq. (<ref>) leads to 


    W_2    = π∫_-r_0^r_0(r_0^2-z^2 )^1/2  z = 12π^2 r_0^2


and the mean radius of curvature from Eq. (<ref>) with 


    R̃_   = 2π r_0∫_-r_0^r_0(r_0^2 - z^2 )^1/2  z = r_0


is again simply the radius of the hypersphere r_0: The radius of curvature of a hypersphere is just the radius of the hypersphere. 

Using the relations in Eq. (<ref>), the quermassintegrals W_i of a four-dimensional hypersphere can be written as

    W_i = 12π^2 r_0^4-i,

agreeing with the general solution for D-dimensional hyperspheres 

    W_i = κ_D r_0^D-i

from the literature <cit.>. 




  §.§.§ Hyperellipsoid


A uniaxial hyperellipsoid of revolution is the affine transformation of a hypersphere with radius r_ in z-direction by a factor ν. 
The meridian curve r(z) of this hyperellipsoid of revolution with the center in the origin of the coordinate system reads as 

    r(z) = [r_^2 - (zν)^2 ]^1/2 .

For the aspect ratio ν =1, a hypersphere results. Hyperellipsoids of revolution with ν >1 are prolate and with ν < 1 are oblate solids of revolution.
The two-dimensional sections of such geometries are shown in Fig. <ref>. 

 


For the volume of a four-dimensional hyperellipsoid of revolution, Eq. (<ref>) leads to 

    V_   = 43π∫_- ν r_^ν r_(r_^2 - z^2ν^2)^3/2  z = 12π^2 ν r_^4

for prolate and oblate hyperellipsoids. 
The total surface area S_ for this geometry is the lateral surface area S_^' which can with Eq. (<ref>) be written as  


    S_   = S_^' = 4 π∫_- ν r_^ν r_(r_^2 - z^2ν^2) [ν^4 r_^2 + z^2 (1-ν^2 )ν^2 (ν^2 r_^2 -z^2 )]^1/2  z 
       = 2 π^2 ν^2 r_^3   _2ℱ_1(52, 12, 2; 1-ν^2 )
 

where _2ℱ_1 is the Gauss hypergeometric function defined as 

    _2ℱ_1 (a,b,c;z ) 
       = Γ(c)Γ(b)Γ(c-b)∫_0^1 t^b-1(1-t )^c-b-1(1-zt )^-a t 
       = ∑_n=0^∞(a )_n (b )_n(c )_nz^nn!

and 

    (q )_n = ∏_k=0^n-1(q+k ) = Γ(q+n)Γ(q)

the Pochhammer symbol. 

Using linear transformation and identities of the hypergeometric function <cit.>, the total surface area can also be denoted as 


    S_ = {[ 8 πν r_^33 (ν^2-1)[(2ν^2-1 ) ℰ(√(ν^2-1)ν)- 𝒦(√(ν^2-1)ν) ]                                                         :                                                     ν > 1;   8 πν r_^33 (1-ν^2)[(1-2ν^2ν) ℰ(√(1-ν^2))+ν𝒦(√(1-ν^2)) ]                                                         :                                                     ν < 1 ].
 

with the complete elliptic integral of the first kind 


    𝒦(k) = π2 _2ℱ_1(12, 12,1;k^2 )


and the complete elliptic integral of the second kind


    ℰ(k) = π2 _2ℱ_1(-12, 12,1;k^2 ) .


For the limiting case of a hypersphere with ν→ 1, for both branches 

    lim_ν→ 1^+ S_(ν) = lim_ν→ 1^- S_(ν) = 2 π^2 r_^3

results which agrees with the expression in Eq. (<ref>). 
For infinitely thin hyperellipsoids of revolution 

    lim_ν→ 0 S_(ν) = 83π r_^3

results. 

For the second quermassintegral W_2 and the mean radius of curvature R̃_, again, the principal radii of curvature R_1(z), R_2(z), and R_3(z) need to be calculated. 
Using Eqs. (<ref>) and (<ref>) 


    R_1(z) = R_2(z) = 1ν^2[ν^4 r_^2 + z^2(1-ν^2 ) ]^1/2

and

    R_3(z) = 1ν^4 r_^2[ν^4 r_^2 + z^2(1-ν^2 ) ]^3/2

are obtained. 
Then the second quermassintegral is, using Eq. (<ref>), 


    W_2    = π3 ν^3∫_-ν r_^ν r_(ν^2 r_^2 - z^2 )^1/2[ν^4 r_^2ν^4 r_^2 + z^2(1-ν^2 ) + 2ν^2 ]   z 
       =13π^2 r_^2 (1-ν^31-ν^2) ,


both for prolate and oblate hyperellipsoids of revolution. In the limit of a four-dimensional hypersphere 


    lim_ν→ 1^+ W_2( ν) = lim_ν→ 1^- W_2( ν) = 12π^2 r_^2


is obtained while 


    lim_ν→ 0 W_2( ν) = 13π^2 r_^2

results for infinitely thin hyperellipsoids of revolution.  Eq. (<ref>) is identical to the expression in Eq. (<ref>). 

Finally, the mean radius of curvature R̃_ for hyperellipsoids of revolution can be written as 


    R̃_   = 23 πν^3 ∫_-ν r_^ν r_(ν^2 r_^2 -z^2 )^1/2{2 ν^6 r_^2 + ν^4 [ν^4 r_^2 + z^2 (1-ν^2 ) ][ν^4 r_^2 + z^2 (1-ν^2 ) ]^3/2}  z 
       = ν^4 r_ _2ℱ_1 (52, 32, 2; 1-ν^2 )


using Eq. (<ref>). 
Using complete elliptic integrals [Eqs. (<ref>) and (<ref>)], it can alternatively be denoted as

    R̃_={[ 43 πν r_ν^2-1[(ν^2-2 )ℰ(√(ν^2-1)ν) +𝒦(√(ν^2-1)ν) ]                                                  :                                              ν > 1; 43 πr_1-ν^2[(2-ν^2 )ℰ(√(1-ν^2)) -ν^2 𝒦(√(1-ν^2)) ]                                                  :                                              ν < 1;                                                    ].

for prolate and oblate hyperellipsoids of revolution. In the limit of a hypersphere,


    lim_ν→ 1^+R̃_( ν) = lim_ν→ 1^-R̃_( ν) = r_

results [Eq. (<ref>)] while for infinitely thin hyperellipsoids 


    lim_ν→ 0R̃_( ν) = 83 π r_


is obtained.

Using Eqs. (<ref>), the general relation

    W_i    = 12π^2 ν^i+1 r_^4-i _2ℱ_1(52,i2, 2; 1-ν^2 ) ,


with


    _2ℱ_1(52,0, 2; 1-ν^2 ) = 1


and 

    _2ℱ_1(52,1, 2; 1-ν^2 ) = 23 ν^3 1-ν^31-ν^2

is obtained in ℝ^4 <cit.>. 
This agrees with the general solution for oblate, D-dimensional ellipsoids of revolution 

    W_i = κ_D ν^i+1 r_^D-i _2ℱ_1(D+12,i2, D2; 1-ν^2 )
 
provided in the literature <cit.>. 
Using the Pfaff transformation <cit.> with


    _2ℱ_1(a,b,c; z ) = (1-z )^-b _2ℱ_1(c-a,b,c; zz-1) ,


the general quermassintegrals can also be reformulated as 


    W_i = κ_D ν r_^D-i  _2ℱ_1(-12,i2, D2; 1-1ν^2)


for prolate, D-dimensional ellipsoids of revolution. 







  §.§.§ Hyperspherocylinder
 

A D-dimensional spherocylinder is the union of a D-dimensional cylinder with height h=2 (ν-1 ) r_ and radius r_, capped by two D-dimensional hemispheres with radius r_. 
A four-dimensional hyperspherocylinder is a prolate solid of revolution without singularities on its surface curvature where the meridian curve r(z) can be written as


    r(z)={[ {r_^2-[z - (ν -1) r_]^2 }^1/2                             :           -(ν-1) r_ < z ≤ν r_;                            r_                             :      -( ν-1) r_≤ z ≤(ν-1 ) r_; {r_^2-[z + (ν -1) r_]^2 }^1/2                             :   (-1) -ν r_≤ z <  -( ν-1) r_;                               ].


for aspect ratios ν≥ 1 with the center of the hyperspherocylinder being in the origin of the coordinate system.
In the limit ν→ 1^+, a hypersphere results. Opposite, in the limit ν→∞, a hard needle in ℝ^4 results.  A two-dimensional section of this geometry is depicted in Fig. <ref>. 

 

With Eq. (<ref>) and its meridian curve [Eq. (<ref>)], the volume of a hyperspherocylinder reads as


    V_   = [12π^2 + 83(ν-1) π] r_^4 .


The total surface area S_ of a hyperspherocylinder, again, is the lateral surface area S_^' 


    S_ = S_^' = [2 π^2 + 8 (ν-1 ) π] r_^3


[Eq. (<ref>)]. 

A hyperspherocylinder's principal radii of curvature [Eqs. (<ref>) and (<ref>) using Eq. (<ref>)] are 


    R_1 = R_2 = r_


and 


    | R_3(z) | = {[                         r_                          :        -(ν-1) r_ < z ≤ν r_;                          ∞                          :   -( ν-1) r_≤ z ≤(ν-1 ) r_;                         r_                          : (-1)-ν r_≤ z <  -( ν-1) r_;                            ].
 

for the cylindrical and hemispherical parts. 
Therefore, the second quermassintegral W_2 reads as 


    W_2 = [ 12π^2 + 43( ν-1 ) π] r_^2
 
using Eq. (<ref>) and, using Eq. (<ref>), the mean radius of curvature as 


    R̃_ = [1+ 43 π(ν-1 ) ] r_ .


Eqs. (<ref>), (<ref>), (<ref>), and (<ref>) agree with the literature <cit.> and, using Eqs. (<ref>), 

    W_i= [12π^2 + 83(ν-1 ) 4-i4π] r_^4-i


is obtained as general expression for the quermassintegrals of a four-dimensional hyperspherocylinder which is in accordance to the relation


    W_i = [ κ_D + 2 (ν-1 ) D-iD κ_D-1] r_^D-i


for D-dimensional spherocylinders as provided in the literature <cit.>.





 §.§ Solids of revolution with singularities on their surface curvature
 

For convex solids of revolution with singularities of their surface curvature, the derivative of the meridian curve ṙ(z) is discontinuous. 
In this case, in addition to the contributions from the piecewise integration over continuous parts  [Eqs. (<ref>), (<ref>), (<ref>), and (<ref>)],
the singularities' contributions have to be considered. As in Sec. <ref>, we describe selected geometries and calculate their 
geometric measures V_, S_, W_2, and R̃_ in the following.



  §.§.§ Spherical hyperplates


The spherical hyperplate as an infinitely thin hyperellipsoid is the limit of a hyperellipsoid with aspect ratio ν→ 0 and radius r_0. 
It is a (D-1)-dimensional sphere with radius r_0 in ℝ^D. 
While in ℝ^3 a hard disk results, in the four-dimensional space a three-dimensional sphere  with a four-dimensional (hyper-)volume  


    V_ = 0


is obtained as analogue to the volumeless disk in ℝ^3. 
This can also be seen from Eq. (<ref>) with a limiting aspect ratio of ν→ 0. 
Despite for this geometry the D-dimensional volume V_ vanishes, the further geometric measures are finite. 
Again, exemplarily the limiting case of a hyperellipsoid with ν→ 0 can be used to obtain


    S_ = 83π r_0^3


for the total surface area of a four-dimensional spherical hyperplate. 
Note, that analogously to a disk in the three-dimensional space the hyperplate has a top and a bottom cell. 
Therefore, the total surface area S_ of a D-dimensional spherical hyperplate is two times the volume of a (D-1)-dimensional sphere
and reads as 


    S_ = 2 κ_D-1 r_0^D-1
 

in Euclidean spaces ℝ^D. The second quermassintegral W_2 for a spherical hyperplate in ℝ^4 is


    W_2 = 13π^2 r_0^2


[Eq. (<ref>)] and its mean radius of curvature

    R̃_ = 83 π r_0


results from Eq. (<ref>). 

Using Eqs. (<ref>), the obtained results agree with the general formulation of quermassintegrals W_i for (D-1)-dimensional spheres in ℝ^D


    W_i = {[                       0                       :                     i=0; κ_D-1κ_i-1iDκ_i r_0^D-i                       :                   i ≥ 1;                         ].


as provided in the literature <cit.>. 





  §.§.§ Hypercylinder




The meridian curve of a hypercylinder with height h=2 ν r_ and radius r_ simply reads as 

    r(z) = r_
 

with vanishing derivatives ṙ(z)=0 and r̈(z)=0. Hence, only R_1=R_2=r_ are finite while R_3 is infinite with vanishing principal curvature 1/R_3.
With ν=h/(2r_) as shown in Fig. <ref>, the volume of a four-dimensional hypercylinder is 


    V_
       = 83πν r_^4


according to Eq. (<ref>). 

Its total surface area S_ is the sum of the lateral surface area 

    S_^'
       = 8 πν r_^3

[Eq. (<ref>)] and the two contributions S_^'' at z=±ν r_ with


    S_^'' = 43π r_^3 .


Hence, the total surface area of a four-dimensional hypercylinder reads as


    S_   = S_^' + 2 S_^'' = 83π(3ν + 1 ) r_^3 .





The contribution of the continuous part W_2^' to the second quermassintegral W_2  is 

    W_2^'
       = 43πν r_^2

[Eq. (<ref>)]. 
For both, top and bottom contribution of the singularity as for the spherical hyperplate 

    W_2^'' = 16π^2 r_^2

results [Eq. (<ref>)]. Hence,


    W_2    = W_2^' + 2 W_2^'' = 13π(4 ν + π) r_^2


is obtained for the second quermassintegral W_2 of a four-dimensional hypercylinder. 

The contribution of the continuous cylindrical part R̃_^' to the mean radius of curvature R̃_ is 


    R̃_^'
       = 43 πν r_


[Eq. (<ref>)] and each contribution of the singularities at z=±ν r_ reads as 


    R̃_^'' = 43 π r_


identical to the top and bottom contribution of the spherical hyperplate [Eq. (<ref>)]. In total,


    R̃_   = R̃_^' + 2 R̃_^'' = 43 π(ν +2 ) r_


results for the mean radius of curvature R̃_ of a four-dimensional hypercylinder.
Using Eqs. (<ref>), the general expression for the quermassintegrals W_i of this geometry 

    W_i = {[                         83πν r_^4                                 :                               i=0; 23π[(4-i ) ν + β_i2 κ_i-1] r_^4-i                                 :                             i ≥ 1;                                   ]. ,


results in accordance to the general result for D-dimensional cylinders 

    W_i = {[                        2 νκ_D-1  r_^D                                     :                                   i=0; 2Dκ_D-1[(D-i ) ν + β_i2 κ_i-1] r_^D-i                                     :                                 i ≥ 1;                                       ].

provided in the literature <cit.>. 





  §.§.§ Hyperspindle


Analogously to a three-dimensional spindle <cit.>, a hyperspindle with center in the origin of the coordinate system can be described by the meridian curve 


    r(z) = (1- ν^22) r_ + 12[(ν^2 +1 )^2 r_^2 - 4z^2 ]^1/2


as a prolate, four-dimensional, uniaxial solid of revolution with radius r_ and aspect ratio ν≥ 1. 



As shown by the two-dimensional section of this geometry in Fig. <ref>, a spindle has two singularities of curvature, located at z=±ν r_ as an upper and lower apex. 
In the limit ν→ 1, the hyperspindle approaches a hypersphere and the contributions of the singularities on the hypersurface vanish. 

The volume of a four-dimensional hyperspindle using Eq. (<ref>) reads as 


    V_
       = 516π r_^4 [(ν^4 - 65ν^2+1 ) (ν^2+1 )^2 arcsin(2νν^2 +1) - . 
           . 2 ν^7 - 1415ν^5 + 1415ν^3 + 2 ν]


in dependence on the radius r_ and the aspect ratio ν. 
The total surface area S_ can be written as 


    S_ = 12π r_^3 [(3 ν^6 + ν^4 + ν^2 +3 ) arcsin(2 νν^2 +1) -6ν^5 +6ν]


using Eq. (<ref>) with Eq. (<ref>) since at the apices r(z_) = r(z_) =0, the zero-dimensional singularities do not contribute to W_1 and S_. 

To calculate the contribution of the apical singularities to W_2 and R̃_, in the interval ν r_ - ϵ≤| z |≤ν r_ the meridian curve r(z) can continuously be replaced by the meridian curve r^♯ (z) of a hypersphere 


    r^♯ (z) = [r_0^2 - (z- ξ)^2 ]^1/2


with 


    r ( ν r_ - ϵ) = r^♯( ν r_ - ϵ)


and 


    ṙ( ν r_ - ϵ) = ṙ^♯( ν r_ - ϵ)


as visualized in Fig. <ref>. 
Here 


    ξ(r_, ν, ϵ) = r_(ν^2-1 ) (ν r_ - ϵ) [(ν^2 +1 )^2 r_^2 - 4 (ν r_ - ϵ)^2 ]^-1/2
 

is the z-position of the hypersphere's center  and 


    r_0 (r_, ν, ϵ) = ν^2+12 r_{1+ (1-ν^2 ) r_[(ν^2+1 )^2 r_^2 - 4 (ν r_ - ϵ)^2 ]^1/2}


its radius. 



In the region continued by a hypersphere, the principal radii of curvature are 


    R_1=R_2=R_3=r_0


[Eqs. (<ref>) and (<ref>)]. 
Using Eq. (<ref>), the second quermassintegral and the mean radius of curvature can, in the vicinity of the apices, be written as 


    W_2^'' (ϵ) = 14∫_0^ϑ_ (ϵ)∫_0^π∫_0^2 π r_0^2 (ϵ) sin^2 ϑsinχ φ χ ϑ


and


    R̃_^'' (ϵ) = 12 π^2∫_0^ϑ_ (ϵ)∫_0^π∫_0^2 π r_0 (ϵ) sin^2 ϑsinχ φ χ ϑ


due to the continuous surface curvature of the hypersphere, using Eqs. (<ref>). 
The critical angle ϑ_ (ϵ) is 


    ϑ_ (ϵ) = arccos[2 (ν r_ - ϵ)(ν^2+1)r_]


(see Fig. <ref>) and performing the integration over φ and χ


    W_2^'' (ϵ) = π∫_0^ϑ_ (ϵ) r_0^2 (ϵ) sin^2 ϑ ϑ


and 


    R̃_^'' (ϵ) = 2π∫_0^ϑ_ (ϵ) r_0 (ϵ) sin^2 ϑ ϑ


result. Hence, in the limit ϵ→ 0  the contributions of the singularities to the second quermassintegral W_2


    W_2^'' = lim_ϵ→ 0 W_2^'' (ϵ) = 0


and the mean radius of curvature R̃_


    R̃_^'' = lim_ϵ→ 0R̃_^'' (ϵ) = 0


vanish. Thus, the second quermassintegral W_2 can be expressed as  


    W_2    = W_2^' + 2 W_2^''
    	 = 512π r_^2 [(ν^4 + 25ν^2 +1 ) arcsin(2 νν^2 +1) - 2 ν^3 + 2 ν]


using Eqs. (<ref>) and (<ref>) with the principal radii of curvature 

    R_1(z) = R_2(z) = ν^2 +12 r_ + (1-ν^2 ) (ν^2 +1 ) r_^22 [(ν^2+1 )^2 r_^2 - 4 z^2 ]^1/2

and 

    R_3 = ν^2 +12 r_


[Eqs. (<ref>) and (<ref>)]. For the mean radius of curvature, finally 

    R̃_   = R̃_^' + 2 R̃_^''
    				= r_π[(ν^2 +1 ) arcsin(2 νν^2 +1) - 23ν ν^2-1ν^2+1]


is obtained [Eqs. (<ref>) and (<ref>)]. 






  §.§.§ Hyperlens


A hyperlens is the section of two hyperspheres with identical radius r_0, separated by the distance L with 0≤ L < 2r_0. 
It is an oblate solid of revolution with equatorial radius r_ and aspect ratio ν with ν∈ [0,1].  
Using equatorial radius r_ and aspect ratio ν, the radius of the intersecting hyperspheres can be written as 


    r_0 = ν^2+12ν r_


and for the distance of separation,  

    L2 = r_0 - ν r_

results. 
Also, since a hyperlens consists of two equal hyperspherical caps, a critical angle θ_ with


    cos(θ_2) = 1-ν^21+ν^2


can be defined, identical to a lens in ℝ^3 <cit.> and a planar lens in ℝ^2 <cit.>. 
In the limit ν→ 1, a hypersphere with radius r_0=r_ results while for the limit ν→ 0, a spherical hyperplate with radius r_ is obtained. 
In Fig. <ref>, the two-dimensional section of a hyperlens is shown with its characteristic parameters. 



Placing the hyperlens' center in the origin of the coordinate system, the meridian curve r(z) reads as


    r(z) = [| z | r_(ν^2-1)+ν(r_^2-z^2)ν]^1/2


where the hyperspheres' centers are located at z=± (r_0 - ν r_). 
Since a hyperlens consists of two attached hyperspherical caps, the principal radii of curvature are with


    R_1 = R_2 = R_3  = ν^2 +12 ν r_ = r_0


equal to the radius of the hyperlens' generating hyperspheres [Eqs. (<ref>) and (<ref>)]. 

For the volume V_ of a hyperlens, 


    V_   = π r_^432 ν^4{(1+ν^2 )^4 [π -2 arcsin(1-ν^21+ν^2) ]-4ν[1-ν^6 + 113ν^2 (1-ν^2 ) ] }


is obtained [Eqs. (<ref>) and (<ref>)]. 

Its total surface area S_ equals the lateral surface area S_^' and can be written as  


    S_   = π(1+ν^2 ) r_^34 ν^3{(1+ν^2 )^2 [π - 2 arcsin(1-ν^21+ν^2) ] - 4 ν(1-ν^2 ) }


using Eq. (<ref>). 

For the contribution of the continuous parts of the hyperlens to its second quermassintegral and mean radius of curvature,


    W_2^'   = π r_^28 ν^2[(1+ν^2 )^2 π - 2 (ν^4+2ν^2+1 ) arcsin(1-ν^21+ν^2) - 4 ν(1-ν^2 ) ]


and 


    R̃_^'   =r_2 πν{(1+ν^2 ) [π - 2 arcsin(1 - ν^21+ν^2) ]-4 ν1-ν^21+ν^2}
 

are obtained [Eqs. (<ref>) and (<ref>) with Eq. (<ref>)]. 

For the contributions of the equatorial singularity W_2^'' and R̃_^'', the surface of a cut-out hyperlens in -ϵ≤ z ≤ϵ can continuously be replaced by a toroidal hypercylinder with height 2 ϵ and meridian curve 


    r^♯ (z) = ξ + (r_^2 -z^2 )^1/2


where 


    r^♯ (ϵ) = r(ϵ)


and 


    ṙ^♯ (ϵ) = ṙ (ϵ)


are fulfilled (see Fig. <ref>). 



In Eq. (<ref>) 


    r_(ϵ) = ϵ  r_(1+ν^2 )2 νϵ + r_(1-ν^2 )


is the meridian radius of the hypertorus and 


    ξ(ϵ) = -2 ϵ [ν(ν r_ - ϵ) ( νϵ + r_) ]^1/22 νϵ + (1-ν^2 ) r_ + [(ν r_ - ϵ) ( νϵ + r_)ν]^1/2


the distance between the center of the hypertorus and the center of the meridian arc. 
In the region of the hypertorus, the principal radii of curvature can be written as 



    R_1(z)    = R_2(z) = r_[1+ ξ(r_^2 - z^2)^1/2]  , 
    
    R_3(z)    = r_



using Eqs. (<ref>) and (<ref>). 
Employing z=r_cosϑ in ϑ_≤ϑ≤π -ϑ_ (see Fig. <ref>) with cosϑ_ (ϵ) = ϵ / r_ (ϵ), the principal radii of curvature can be written as 



    R_1(ϑ)    = R_2(ϑ) = r_(ϵ) + ξ(ϵ)sinϑ , 
    
    R_3(ϑ)    = r_(ϵ)



in dependence on the angular coordinate ϑ. 
With Eq. (<ref>), the second quermassintegral and the mean radius of curvature can, in the vicinity of the equatorial singularity, be written as 


    W_2^'' (ϵ)    = π3∫_ϑ_(ϵ)^π-ϑ_(ϵ)[R_1(ϑ) R_2(ϑ) + R_1(ϑ)R_3(ϑ) + R_2(ϑ)R_3(ϑ) ] sin^2 ϑ ϑ
       = π3∫_ϑ_(ϵ)^π-ϑ_(ϵ)[3 r_^2(ϵ) sin^2 ϑ + 4 r_(ϵ) ξ(ϵ) sinϑ + ξ^2(ϵ)]  ϑ


and


    R̃_^'' (ϵ)    = 23π∫_ϑ_(ϵ)^π-ϑ_(ϵ)[R_1(ϑ) + R_2(ϑ) + R_3(ϑ) ] sin^2 ϑ ϑ
       = 2π∫_ϑ_(ϵ)^π-ϑ_(ϵ)[r_(ϵ) sin^2 ϑ + 23ξ(ϵ) sinϑ]   ϑ


due to the continuous surface curvature of the hypertorus [Eqs. (<ref>)]. 

The contributions of the equatorial singularity 


    W_2^'' = lim_ϵ→ 0 W_2^'' (ϵ) = 23π r_^2 arcsin(1-ν^21+ν^2)


and 


    R̃_^''   = lim_ϵ→ 0R̃_^'' (ϵ)  = 83π r_ 1-ν^21+ν^2


result in the limit ϵ→ 0. 
Hence, for the second quermassintegral and the mean radius of curvature, the contributions 


    W_2^'' = 13π^2 r_^2  2πarcsin[ cos(θ_2) ]


and 


    R̃_^'' = 83π r_ cos(θ_2)


result depending on the critical angle θ_ [Eq. (<ref>)]. In the limit of a spherical hyperplate with θ_=0, Eqs. (<ref>) and (<ref>)
are consistently obtained. Finally, 


    W_2 = W_2^' + W_2^''   =π r_^28 ν^2[(1+ν^2)^2 π -2 (ν^4-23ν^2 +1 ) arcsin(1-ν^21+ν^2) -  . 
         . 4 ν(1-ν^2 ) ]


and 


    R̃_   = R̃_^' + R̃_^'' = r_2 πν{(1+ν^2 ) [π - 2 arcsin(1 - ν^21+ν^2) ]+43ν1-ν^21+ν^2}


are obtained for the second quermassintgral W_2 and mean radius of curvature R̃_ of a hyperlens.




Additionally, in the limit ν→ 0, the geometric measures S_, W_2, and R̃_ with Eqs. (<ref>), (<ref>), and (<ref>) can, for numerical stability, be expanded in Maclaurin series as


    S_   = 83π r_^3 [1+ 65ν^2 + 𝒪(ν^4) ]  ,




    W_2    = 13π r_^2 [π + 3215ν^3 + 𝒪(ν^5) ]  ,


and 


    R̃_   = 83 π r_[1+25ν^4 + 𝒪(ν^6)]


to obtain the limits of a spherical hyperplate with Eqs. (<ref>), (<ref>), and (<ref>). 
For the volume V_ [Eq. (<ref>)], the expansion 


    V_   = 16 π r_^4 [115ν + 135ν^3 + 𝒪(ν^5)]


is obtained. 







  §.§.§ Hyperspherical cap


A hyperspherical cap is a hypersphere cut off by a single hyperplane resulting in half of a spherical hyperplate as a singularity on the surface of this geometry. 
The hyperspherical cap can be defined by the radius of the hypersphere r_0 and the critical angle θ_ (Fig. <ref>). 
With these parameters, the height  

    h = r_0 [1-cos(θ_2) ]

and radius of the cap 

    r_ cap = r_0 sin(θ_2) = [h ( 2 r_0 -h ) ]^1/2
 

are obtained, resulting in the aspect ratio 


    ν = {[     h2 r_         :   0 ≤θ_≤π;          ;    h2 r_0         : π≤θ_≤ 2 π ].
 

of hyperspherical caps. In the limit ν→ 1, a hypersphere with radius r_0 results while for ν→ 0, a spherical hyperplate with radius r_ is obtained. 
Placing the hypersphere's center in the origin of the coordinate system, the meridian curve r(z) of the hyperspherical cap 


    r(z) = (r_0^2 -z^2 )^1/2


is identical to the meridian curve of the complete hypersphere [Eq. (<ref>)]. 


 

For the volume of a hyperspherical cap Eq. (<ref>) can be used analogously to Eq. (<ref>) as 

    V_   = 43π∫_r_0-h^r_0(r_0^2 -z^2 )^3/2 z 
       = π^24 r_0^4 [1- 2πarctan(r_0 -hr_) - 2 r_3 π r_0^4(r_0-h ) (3r_0^2 +2 r_^2 ) ]

by replacing the limits of the integral accordingly.   

This agrees with 


    V_ = κ_D2 r_0^D ℐ_r_ cap^2/r_0^2(D+12,12)      :    0 ≤ h ≤ r_0


as the expression for the volume of D-dimensional spherical caps in ℝ^D with θ_≤π provided in the literature <cit.>. 
Here ℐ_x(a,b) with


    ℐ_x(a,b)    = Γ(a+b)Γ(a) Γ(b)∫_0^x t^a-1 (1-t)^b-1  t  
       = Γ(a+b)Γ(a) Γ(b)x^aa _2ℱ_1 (a,1-b,a+1;x )


is the regularized incomplete beta function <cit.>. 
Also Eq. (<ref>) can easily be adapted to 


    V_ = κ_D2 r_0^D [ 2-  ℐ_r_^2/r_0^2(D+12,12)]      :    r_0 ≤ h ≤ 2r_0


for hyperspherical caps with critical angles θ_≥π while, in ℝ^4, Eq. (<ref>) can be used for the volume of hyperspherical caps with arbitrary critical angles θ_. 


Its total surface area S_ can be written as the sum


    S_ = S_ ^' + S_^''


of the lateral surface area S_^' and the surface area at the singularity S_^''. 
For the lateral surface area S_^', Eqs. (<ref>) and (<ref>) can be used to obtain


    S_^'
       = π^2 r_0^3 [1-2πarctan(r_0-hr_) - 2 r_π r_0^2(r_0 -h ) ]
 

while the surface area at the singularity 


    S_^'' = 43π r_^3


is half the surface area of a two-sided spherical hyperplate. 
Hence, the surface area of hyperspherical caps reads as


    S_ = π^2 r_0^3 [1-2πarctan(r_0-hr_) - 2 r_π r_0^2(r_0 -h ) ] + 43π r_^3


in ℝ^4.


For the lateral surface area S_^' of D-dimensional spherical caps with θ_≤π

    S_^' = β_D2  r_0^D-1ℐ_r_^2/r_0^2(D-12,12)       :    0 ≤ h ≤ r_0

is available in the literature <cit.> which can be adapted to  

    S_^' = β_D2  r_0^D-1[2- ℐ_r_^2/r_0^2(D-12,12)]       :    r_0 ≤ h ≤ 2 r_0

for θ_ > π. In ℝ^4, Eq. (<ref>) can be used for the lateral surface area of hyperspherical caps with arbitrary critical angles θ_. 
Using 


    S_^'' = κ_D-1 r_^D-1


for the generalized contribution of the singularity, the expression 

    S_ = {[      β_D2  r_0^D-1ℐ_r_^2/r_0^2(D-12,12) + κ_D-1 r_^D-1                                                      :                                            0 ≤ h ≤ r_0; β_D2  r_0^D-1[2- ℐ_r_^2/r_0^2(D-12,12)] + κ_D-1 r_^D-1                                                      :                                        r_0 ≤ h ≤ 2 r_0 ].


results for the total surface area of D-dimensional spherical caps in ℝ^D. 


For the second quermassintegral 

    W_2 = W_2^' + W_2^'' ,

the contribution for the continuous part W_2^' can be written as 

    W_2^'
       = π^24 r_0^2 [1-2πarctan(r_0-hr_) - 2 r_π r_0^2(r_0 -h ) ]

using Eq. (<ref>) with the principal radii of curvature  

    R_1 = R_2 = R_3 = r_0

resulting from the meridian curve r(z) [Eq. (<ref>)]. 
Using the regularized incomplete beta function ℐ_x(a,b), also 


    W_2^' = {[      14π^2 r_0^2ℐ_r_^2/r_0^2(32,12)                                   :                         0 ≤ h ≤ r_0; 14π^2 r_0^2[ 2-ℐ_r_^2/r_0^2(32,12)]                                   :                     r_0 ≤ h ≤ 2 r_0 ].


can be formulated in analogy to the previously described geometric measures. 
The contribution of the singularity W_2^'' consists of half the contribution of a spherical hyperplate with radius r_ [Eq. (<ref>)] and half the contribution of the singularity of a hyperlens with radius r_ [Eq. (<ref>)] where


    W_2^''   = 12 13π^2 r_^2 + 12 23π r_^2 arcsin[ cos(θ_2) ] 
       = 13π r_^2  [π2+ arcsin( 1- hr_0)]


is obtained. 
Using Eqs. (<ref>), (<ref>), and (<ref>)


    W_2    = π^24 r_0^2 [1-2πarctan(r_0-hr_) - 2 r_π r_0^2(r_0 -h ) ]
            + π3  r_^2  [π2 +  arcsin(1-hr_0) ]


results for the second quermassintegral W_2 of hyperspherical caps in ℝ^4. 

For the mean radius of curvature 


    R̃_ = R̃_^' + R̃_^'' ,


the contribution of the continuous part R̃_^' is 


    R̃_^'
       = r_02[1-2πarctan(r_0-hr_) - 2 r_π r_0^2(r_0 -h ) ]


using Eqs. (<ref>) and (<ref>). 
Using the regularized incomplete beta function ℐ_x(a,b), R̃^'_ can be reformulated as 


    R̃_^' = {[     r_02 ℐ_r_^2/r_0^2(32,12)                            :                  0 ≤ h ≤ r_0; r_02[ 2-ℐ_r_^2/r_0^2(32,12)]                            :              r_0 ≤ h ≤ 2 r_0 ]. .


For the contribution at the singularity R̃_^'', again, half the contribution of a spherical hyperplate [Eq. (<ref>)] and half the contribution of the singularity of a hyperlens [Eq. (<ref>)] are added as


    R̃_^''   = 12 83π r_ + 12 83 π r_cos(θ_2) = 43 π r_(2-hr_0)


where


    R̃_ = r_02[1-2πarctan(r_0-hr_) - 2 r_π r_0^2(r_0 -h ) ] + 43 π r_(2-hr_0)


results for the total mean radius of curvature R̃_. 






  §.§.§ Hyperdoublecone
 

A hyperdoublecone is a uniaxial solid of revolution with inversion symmetry whose meridian curve can be written as


    r(z) = r_ - | z |ν


in dependence on the equatorial radius r_ and aspect ratio ν.
Hyperdoublecones exist as oblate (ν≤ 1) and prolate (ν≥ 1) solids with the limiting cases of a spherical hyperplate
for ν→ 0 and a hard needle for ν→∞ in ℝ^4 analogously to three-dimensional doublecones <cit.>. 
The two-dimensional section of a hyperdoublecone with aspect ratio ν=2 is shown in Fig. <ref>. 



This geometry has an equatorial singularity with a critical angle θ_ which is defined by 


    cos( θ_) = 1√(1+ν^2) ,


similar to hyperlenses. 
Additionally, two apical singularities exist at z=±ν r_ similar to hyperspindles. 



The volume V_ can be written as 


    V_   = 43π∫_- ν r_^ν r_(r_ - | z |ν)^3   z = 23πν r_^4


using Eqs. (<ref>) and (<ref>). 
Since the lateral surface area S_^' of this geometry equals its total surface area S_, Eq. (<ref>) can be used and 

    S_ = S_^' = 4 π√(1+ν^2)ν∫_-ν r_^ν r_(r_ - | z |ν)^2    z = 83π r_^3 √(1+ν^2)

is obtained.

For the principal radii of curvature 


    R_1(z) = R_2(z)= (r_ - | z |ν) √(1+ν^2)ν


and 


    | R_3| = ∞


result from Eqs. (<ref>) and (<ref>) using Eq. (<ref>).

For the second quermassintegral the contribution of the continuous part W_2^' can be written as 


    W_2^' = 23π∫_-ν r_^ν r_(r_ - | z |ν)  z = 23πν r_^2


[Eq. (<ref>)]. 
For the contribution of the equatorial singularity W_2^'', analogously to hyperlenses, an infinitely thin hypertorus can be used for a continuous completion. 
Therefore, 


    W_2^''   = 23π r_^2 arcsin[cos(θ_)  ] = 23π r_^2 arcsin( 1√(1+ν^2))
 

results, similar to hyperlenses [Eq. (<ref>)]. 

For the contribution at the apical singularities W_2^''', similar to hyperspindles, 


    W_2^''' = 0


is obtained [Eq. (<ref>)]. 
Therefore, the second quermassintegral can be written as 


    W_2    = W_2^' + W_2^'' + 2 W_2^''' = 23π r_^2 [ν + arcsin( 1√(1+ν^2))  ]


in dependence on the equatorial radius r_ and the aspect ratio ν. 

For the mean radius of curvature R̃_, similar to the second quermassintegral W_2,  


    R̃_^'
    = 43 π  r_ν^2√(1+ν^2)


is obtained from Eq. (<ref>) for the contribution of the continuous part R̃_^', 


    R̃_^'' = 83 π r_cos( θ_) = 83 π  r_1√(1+ν^2)


for the contribution of the equatorial singularity R̃_^'' [Eq. (<ref>)], and 


    R̃_^''' = 0


for the contribution of the apical singularities R̃_^''' [Eq. (<ref>)]. 
Hence, 


    R̃_   = R̃_^'  + R̃_^''  +2 R̃_^''' = 43 π r_ν^2 +2√(1+ν^2)


results for the mean radius of curvature R̃_.  





  §.§.§ Hypercone


Similar to a hyperdoublecone, placing the spherical base's center in the coordinate system's origin, the meridian curve of a hypercone reads as


    r(z) = r_0 - zν


in dependence on its base radius r_0 and aspect ratio ν.
In Fig. <ref>, the two-dimensional section of a hypercone with aspect ratio ν =2 is shown. 

Identical to the hyperdoublecone, a hypercone's critical angle θ_ is 


    cos( θ_) = 1√(1+ν^2) .




The volume of a four-dimensional hypercone reads as 


    V_ = 43π∫_0^ν r_0(r_0 - zν)^3   z 
    = 13πν r_0^4


[Eq. (<ref>)], half the volume of a corresponding hyperdoublecone [Eq. (<ref>)]. 

Its lateral surface area S_^' can be written as 


    S_^'
    = 43π r_0^3 √(1+ν^2)


[Eq. (<ref>)], half the total surface area of the corresponding hyperdoublecone [Eq. (<ref>)]. With the contribution of the base 


    S_^'' = 43π r_0^3 ,


the total surface area reads as


    S_   = S_^' + S_^'' = 43π r_0^3 ( √(1+ν^2)  +1 )


in dependence on the radius r_0 and aspect ratio ν. 
For the contribution of the continuous part to the second quermassintegral


    W_2^'
    = 13πν r_0^2


[Eq. (<ref>)], half of the contribution of a corresponding hyperdoublecone is obtained [Eq. (<ref>)]. 
The principal radii of curvature with 


    R_1(z) = R_2(z) = (r_0 - zν) √(1+ν^2)ν
 

and


    | R_3| =  ∞
 

of a hypercone are identical to those of a hyperdoublecone. Similar to hyperspherical caps, the contribution of the singularity at the base W_2^'' consists of 
half the contribution of a spherical hyperplate and half the contribution of a hyperdoublecone's singularity.
Hence, the base singularity contributes 

    W_2^''   = 16π^2 r_0^2 + 13π r_0^2 arcsin[cos(θ_) ] 
       = 16π r_0^2 [ π + 2 arcsin(1√(1+ν^2)) ]


to the second quermassintegral. Since the contribution of the apical singularity W_2^'''=0 vanishes, the second quermassintegral W_2 of a hypercone can be written as 


    W_2    = W_2^' + W_2^'' + W_2^''' = 16π r_0^2 [2ν +π + 2arcsin(1√(1+ν^2)) ] .



Similar, the contribution R̃_^' of the continuous surface curvature to the mean radius of curvature  


    R̃_^'
    =23 π r_0 ν^2√(1+ν^2)


is half the contribution of a corresponding hyperdoublecone [Eq. (<ref>)]. 
For the contribution at the base singularity, analogously to the contribution of the second quermassintegral, half a spherical hyperplate and half a hyperdoublecone can be used, resulting in 


    R̃_^''   = 12 83 π r_0 + 12 83 π r_0 cos(θ_)  = 43 π r_0 ( 1 + 1√(1+ν^2)) .


Since the apical singularity with R̃_^''' = 0 [Eq. (<ref>)] does not contribute to the mean radius of curvature, R̃_ 
of a four-dimensional hypercone can be written as


    R̃_   = R̃_^' + R̃_^'' + R̃_^''' = 23 π r_0 (2+ν^2+2√(1+ν^2)) .






  §.§.§ Truncated hypercone
 

When a hypercone with base radius r_0 and aspect ratio ν is cut off by a hyperplane parallel to the base at height h, a truncated hypercone results with radius r_1 at the second singularity. 
The section of such a truncated hypercone is shown for an aspect ratio of ν =2 and height h=r_0 in Fig. <ref>. 
The meridian curve r(z) is again 


    r(z) = r_0 - zν = r_0 - r_0 -r_1h  z


with the center of the base with radius r_0 as origin of the coordinate system. 
The aspect ratio ν can be defined by 


    ν = hr_0 -r_1 = tan( θ_)


restricting r_1 < r_0. The case r_1=r_0 resulting in a hypercylinder with aspect ratio ν^'=h/(2 r_0) is discussed in section <ref>.
In the limits ν→ 0 and h → 0 a spherical hyperplate with radius r_0 =r_1 results while for the limit r_1 / r_0 →  0 a hypercone with radius r_0 and aspect ratio ν is obtained. 

Again, θ_ is the critical angle with 


    cos(θ_) = 1√(1+ν^2) ,


identical to hyperdoublecones and hypercones. 

 

The volume V_ of a truncated hypercone can with Eq. (<ref>) be written as 

    V_   = 43π∫_0^h (r_0 - r_0 -r_1h z )^3  z = h3π(r_0^3 + r_0^2r_1 + r_0r_1^2 + r_1^3 )


which can be extended to 


    V_ = κ_D-1hDr_1^D - r_0^Dr_1 -r_0


for D-dimensional truncated cones in Euclidean spaces ℝ^D. 

Its lateral surface area S_^' can be written as


    S_^'
       = 43π(r_0^2 + r_0 r_1 + r_1^2 ) √(h^2+(r_0-r_1)^2)


[Eq. (<ref>)] and with the contributions  

    S_^'' = 43π r_0^3


and 


    S_^''' = 43π r_1^3


at the singularities, the total surface area S_=S_^' + S_^'' + S_^''' as


    S_ = 43π[ r_0^3 + r_1^3 + (r_0^2 + r_0 r_1 + r_1^2 ) √(h^2 + (r_0 -r_1 )^2) ]


is obtained. This can be generalized to ℝ^D as 

    S_ = κ_D-1[r_0^D-1 + r_1^D-1 + r_1^D-1-r_0^D-1r_1-r_0√(h^2 + (r_0 -r_1 )^2) ]


using the common relation

    κ_D = β_DD


of the volume κ_D to the surface area β_D for a D-dimensional unit sphere [Eqs. (<ref>) and (<ref>)]. 
Combining the meridian curve [Eq. (<ref>)] with Eqs. (<ref>) and (<ref>) 


    R_1(z) = R_2(z)    = (r_0 - r_0 -r_1h z ) √(h^2 + ( r_0 -r_1)^2)h
 

and


    | R_3| =  ∞
 

result as principal radii of curvature.
Herewith, the contribution of the part with continuous surface curvature to the second quermassintegral of a truncated hypercone can be written as


    W_2^'   = 13π h (r_0 + r_1 )


using Eq. (<ref>). 
For the contribution at the base singularity, similar to hypercones


    W_2^''   = 12 13π^2 r_0^2 + 12 23π r_0^2 arcsin[cos(θ_) ] 
       = 16π r_0^2 [π + 2 arcsin(r_0 -r_1√(h^2+ ( r_0 - r_1 )^2)) ]


results, using half of a spherical hyperplate and half of a hyperdoublecone. 
For the contribution at the top singularity to the second quermassintegral 


    W_2^'''   = 12 13π^2 r_1^2 - 12 23π r_1^2 arcsin[cos(θ_) ] 
       = 16π r_1^2 [ π- 2arcsin(r_0-r_1√(h^2+(r_0-r_1)^2))]


is obtained in analogy to three-dimensional truncated cones <cit.>. 
In total,  


    W_2 = W_2^' + W_2^'' + W_2^''' =   13π(r_0+r_1 ) {h + (r_0 -r_1 ) arcsin[r_0-r_1√(h^2+(r_0-r_1)^2)] }
          +16π^2 (r_0^2+r_1^2 )


results as second quermassintegral W_2 for truncated hypercones in ℝ^4. 

Finally, the mean radius of curvature R̃_ of a truncated hypercone consists of the contribution of the continuous part 


    R̃_^' = 23πh^2√(h^2 + (r_0-r_1)^2)


[Eq. (<ref>)] and the contributions of both singularities    


    R̃_^''   = 12 83 π r_0 + 12 83 π r_0 cos(θ_) = 43 π r_0 (1+ r_0-r_1√(h^2+(r_0-r_1)^2))


and


    R̃_^'''   = 12 83 π r_1 - 12 83 π r_1 cos(θ_) = 43 π r_1 (1-r_0-r_1√(h^2+(r_0-r_1)^2)) .
 

Hence, for the mean radius of curvature of a truncated hypercone 


    R̃_   = R̃_^' + R̃_^'' +R̃_^''' = 23 π[2 (r_0+r_1) + h^2 + 2 (r_0-r_1)^2√(h^2+(r_0-r_1)^2)]


is obtained.






§ RESULTS


Using the geometric measures volume V_, total surface area S_, second quermassintegral W_2, and mean radius of curvature R̃_, the second virial coefficient B_2, representing the excluded volume per particle for the four-dimensional, convex solids of revolution provided in Sec. <ref>, can be obtained using Eq. (<ref>). 
To investigate the influence of the specific particle shape in dependence on the aspect ratio ν, these virial coefficients can be normalized to the particle volume V_ using 


    B_2^∗ = B_2V_ = 1+ S_R̃_V_ + 6π^2W_2^2V_


as reduced second virial coefficients B_2^∗. 
The resulting reduced virial coefficients, independent of the particle size, allow an expansion of the real gas factor Z in powers of the packing fraction η [Eq. (<ref>)] 
and  thus a comparison of different geometric shapes.




 §.§ Solids with inversion symmetry
 

In Fig. <ref>, the second virial coefficients B_2 of  hyperellipsoids, hyperspherocylinders, hyperspindles, hypercylinders, hyperdoublecones, and hyperlenses as four-dimensional, uniaxial solids of revolution with inversion symmetry are shown. While hyperspherocylinders and hyperspindles are strictly prolate geometries with ν≥ 1,  hyperlenses are strictly oblate geometries with 0 ≤ν≤ 1.  
Hyperellipsoids, hypercylinders, and hyperdoublecones can be prolate or oblate with 0≤ν < ∞. 




As visible in Fig. <ref>, hyperellipsoids, hyperspherocylinders, hyperspindles, and hyperlenses approach in limit ν→ 1  hyperspheres with radius r_ and 


    B_2^ ( hsph )  = 4 π^2 r_^4


as their second virial coefficient. 

For oblate geometries, with decreasing aspect ratio all geometries approach in the limit 


    lim_ν→ 0B_2^(hdcn)r_^4   = 
    lim_ν→ 0B_2^(hlen)r_^4 =
    lim_ν→ 0B_2^(hell)r_^4 =
    lim_ν→ 0B_2^(hcyl)r_^4 
    = 649 + 23π^2


spherical hyperplates with radius r_ and 

    B_2^(shyp) = (649 + 23π^2 ) r_^4

as their second virial coefficient. 


In the limit ν→∞, hard hyperneedles result where hyperspherocylinders approach hypercylinders since the contribution of the capping hyperspheres becomes negligible. 
For sufficiently large aspect rations, the proportionality B_2(ν≫ 1) ∝ν^2 is observed as indicated by the black line 5ν^2. 
The limits of B_2/(ν^2 r_^4) with 



    lim_ν→∞B_2^(hcyl)ν^2 r_^4   = 643 , 
    lim_ν→∞B_2^(hscy)ν^2 r_^4   = 643 , 
    lim_ν→∞B_2^(hell)ν^2 r_^4   =  649 + 23π^2  , 
    lim_ν→∞B_2^(hspi)ν^2 r_^4   = 1408135 , 
    lim_ν→∞B_2^(hdcn)ν^2 r_^4   = 569



decrease from hypercylinders (hcyl) and hyperspherocylinders (hscy) with identical value over hyperellipsoids (hell) and hyperspindles (hspi) to hyperdoublecones (hdcn).
This sequence is in accordance to the relation 


    K^(hdcn)⊂ K^(hspi)⊂ K^(hell)⊂ K^(hscy)⊂ K^(hcyl)


for convex sets K≡ K(ν,r_) of hyperdoublecones, hyperspindles, hyperellipsoids, hyperspherocylinders, and hypercylinders with identical aspect ratio ν and equatorial radius r_.
This behavior is identical to their three-dimensional analogues <cit.>. 




  §.§.§ Prolate geometries


To analyze the influence of the detailed particle geometry, reduced virial coefficients B_2^∗ are compared in Fig. <ref> (lhs) in dependence on their aspect ratio ν. 
For a better display of relative deviances, the ratios of reduced second virial coefficients B_2^∗(ν) to the reduced second virial coefficient B_2^∗  (hell)(ν) of hyperellipsoids are shown in Fig. <ref> (rhs).  
In the limit ν→ 1, hyperellipsoids, hyperspherocylinders, and hyperspindles approach hyperspheres with a reduced second virial coefficient of B_2^∗ (hsph) = 8. 
For arbitrary dimensions D, the reduced second virial coefficient of D-dimensional spheres is 


    B_2^∗ (hsph) = 2^D-1


as commonly known <cit.>. 




With increasing aspect ratio ν, the proportionality B_2^∗ (ν≫ 1) ∝ν is observed with the limits 



    lim_ν→∞B_2^∗ (hcyl)ν   = 8π ,
    lim_ν→∞B_2^∗ (hscy)ν   = 8π ,
    lim_ν→∞B_2^∗ (hspi)ν   = 779 π ,
    lim_ν→∞B_2^∗ (hell)ν   = 43 + 1289 π^2 , 
    lim_ν→∞B_2^∗ (hdcn)ν   = 283π



in increasing order [Fig. <ref> (rhs)].
Again, for large aspect ratios hyperspherocylinders approach hypercylinders with a negligible contribution of the capping hyperspheres. 
The limit for four-dimensional hyperspherocylinders is already reported in a previous work <cit.>. 

Particularly interesting is the limiting case ν→ 1 for hypercylinders and hyperdoublecones: 
For four-dimensional hypercylinders and hyperdoublecones with aspect ratio ν=1, the reduced second virial coefficient 


    [B_2^∗ (hcyl)(ν=1)] = [B_2^∗ (hdcn)(ν=1)] = 3 + 20π + π4


is identical and thus the rotation-averaged excluded volume V_ for particles with the same volume V_ is identical, too.  

For the volumes of D-dimensional cylinders and doublecones in ℝ^D, in general


    V_^(cyl)(ν) = D V_^(dcn)(ν) = 2  κ_D-1ν r_^D


is obtained [Eqs. (<ref>), (<ref>), and (<ref>)]. 
The reduced second virial coefficients B_2^∗ of four-dimensional hypercylinders and hyperdoublecones can also be compared to B_2^∗ of their analogue geometries in lower-dimensional Euclidean spaces: 
In ℝ^2, for the meridian curve of a hypercylinder a rectangle and for the meridian curve of a hyperdoublecone a rhombus with half the area of the corresponding rectangle are obtained.
In the limit ν→ 1, hard squares result for both geometries and therefore also the same reduced second virial coefficient B_2^∗ = 1 + 4 / π results <cit.>. 

In ℝ^3, however, the reduced second virial coefficients of cylinders and doublecones with aspect ratio ν=1 differ <cit.>. 
This results lead to the conjecture that the rotation-averaged excluded volumes of D-dimensional, uniaxial cylinders and doublecones with aspect ratio ν=1 and identical volumes V_^ (cyl)=V_^ (dcn) are identical in even-dimensional Euclidean spaces and different in odd-dimensional spaces. The proof is beyond the scope of this work and will be topic of future work. 


Comparing Fig. <ref> (rhs) with the analogue three-dimensional results  (Fig. 7 in <cit.>), characteristic differences are observed between ℝ^3 and ℝ^4:  
For large aspect ratios ν≫ 1, the ratios of the reduced second virial coefficients B_2^∗/B_2^∗ ( hell) are more similar in ℝ^4 than in ℝ^3 due to the additional symmetry of the solid. 
The reduced second virial coefficients of hyperspherocylinders and hypercylinders in ℝ^4 normalized to those of hyperellipsoids similarly depend on the aspect ratio ν  as the reduced second virial coefficients of spherocylinders and cylinders normalized to ellipsoids in ℝ^3. However, for solids with apical singularities, deviations are observed. 
Opposite to spindles in ℝ^3, for hyperspindles in ℝ^4 the ratio [B_2^∗(ν)/B_2^∗ ( hell)(ν)] <1 results for ν→∞. 
Also for hyperdoublecones with increasing aspect ratio ν, the ratio B_2^∗(ν)/B_2^∗ ( hell)(ν) is significantly smaller than for doublecones in ℝ^3. Especially, the reduced second virial coefficients of geometries with apical singularities differ much less in ℝ^4 at large aspect ratios ν≫ 1 from those without apical singularities  than in ℝ^3.




  §.§.§ Oblate geometries


While hyperellipsoids of revolution, hypercylinders, and hyperdoublecones exist in addition to prolate aspect ratios ν≥ 1 also as oblate solids of revolution with 0≤ν≤ 1, hyperlenses are restricted to oblate aspect ratios. In Fig. <ref> (lhs), reduced second virial coefficients B_2^∗ of these geometries are shown in dependence on their inverse aspect ratio ν^-1. 



In the limit ν^-1→ 1, hyperlenses and hyperellipsoids approach hyperspheres with B_2^∗ =8. 
With decreasing aspect ratio ν, the proportionality  B_2^∗(ν≪ 1) ∝ν^-1 is observed with the limits 



    lim_ν→ 0[ν B_2^∗ (hcyl)]    = π4 + 83 π ,
    lim_ν→ 0[ν B_2^∗ (hell)]    = 43 + 1289 π^2 , 
    lim_ν→ 0[ν B_2^∗ (hlen)]    = 58π + 203 π ,
    lim_ν→ 0[ν B_2^∗ (hdcn)]    = π + 323 π



in increasing order. Note that for hyperellipsoids of revolution the relation


    lim_ν→ 0[ν B_2^∗(ν )] = lim_ν→∞B_2^∗(ν )ν


is fulfilled [Eqs. (<ref>) and (<ref>)].

Despite hyperlenses and oblate hyperellipsoids are similar particle shapes, the reduced second virial coefficients of hyperlenses exceed those of the corresponding hyperellipsoids of revolution for all aspect ratios ν < 1, indicating the influence of the detailed particle shape beyond size and aspect ratio.
In ℝ^4, for small aspect ratios 0<ν≪ 1, larger differences between the reduced second virial coefficients of these geometries are observed than between their analogues in ℝ^3 [Fig. 11 in <cit.> and Fig. <ref> (rhs)] which is opposite to the comparison of prolate geometries between ℝ^3 and ℝ^4. 





  §.§.§ Comparison of shapes existing both as prolate and oblate solids of revolutions


Hyperellipsoids, hypercylinders, and hyperdoublecones exist both as prolate and oblate solids of revolution with aspect ratios . Since in addition to the former geometries with inversion symmetry also hypercones (discussed later in Sec. <ref>) exist as prolate and oblate solids of revolution, they are for completeness included in this comparison.
In Fig. <ref>, reduced second virial coefficients B_2^∗ of these geometries are displayed as well as reduced second virial coefficients normalized to those of hyperellipsoids of revolution in dependence of their aspect ratio ν.

 

The reduced second virial coefficient B_2^∗ for each geometry traverses a minimum B_2, ^∗ at ν_ min as summarized in Table <ref>. 
As expected, hyperellipsoids approach for ν→ 1 a hypersphere with the smallest B_2^∗ possible while for different geometries the minima are located at ν_min≠ 1. 


 

In addition to the symmetry relation for reduced second virial coefficients of hyperellipsoids of revolution in the limits ν→ 0 and ν→∞ [Eq. (<ref>)], a general
symmetry relation 


    B_2^∗  (hell)(ν) = B_2^∗  (hell) (ν^-1)


can be identified in ℝ^4 which also exists for ellipsoids of revolution in ℝ^3 <cit.>. This symmetry relation exists for D-dimensional ellipsoids of revolution in arbitrary dimensional Euclidean spaces ℝ^D as proven in the following.

For D-dimensional, uniaxial ellipsoids of revolution with aspect ratio ν, the relation

    B_2^∗ (ell) (D, ν)    = B_2^∗ (ell) (D, ν^-1)

is fulfilled for the reduced second virial coefficient B_2^∗ (ell) in Euclidean spaces ℝ^D with arbitrary dimension D. 


Starting from the Brunn-Minkowski theorem


    V_(K) = 1κ_D∑_i=0^D [ D; i;   ]  W_i(K) W_D-i(K)


[Eq. (<ref>)] for the rotation-averaged excluded volume of two identical convex particles K, their reduced second virial coefficient B_2^∗ (K) can be written as 

    B_2^∗ (K) = 12 W_0(K) κ_D∑_i=0^D [ D; i;   ]  W_i(K) W_D-i(K)


with B_2^∗ = B_2 / V_ [Eq. (<ref>)], V_(K) = 2 B_2(K) [Eq. (<ref>)], and V_(K) = W_0(K) [Eq. (<ref>)]. 
For D-dimensional, uniaxial ellipsoids of revolution, the quermassintegral W_i can be determined by


    W_i = κ_D ν^i+1 r_^D-i _2ℱ_1(D+12,i2,D2;1-ν^2 )


[Eq. (<ref>)]. 
With 


    W_0 = κ_D ν r_^D ,


the reduced second virial coefficient B_2^∗ for a D-dimensional, uniaxial ellipsoid of revolution reads as 


    B_2^∗ (ell) (D, ν) = 12κ_D^2 ν r_^D   ∑_i=0^D [ [ D; i;   ]κ_D^2 ν^D+2 r_^D  _2ℱ_1 (D+12,i2,D2;1-ν^2 ) . 
           .  _2ℱ_1 (D+12,D-i2,D2;1-ν^2 ) ]


which can be rewritten as


    B_2^∗ (ell) (D, ν) = 12ν^D+1   ∑_i=0^D [ [ D; i;   ] _2ℱ_1 (D+12,i2,D2;1-ν^2 ) . 
           .  _2ℱ_1 (D+12,D-i2,D2;1-ν^2 ) ]  .


Analogously, for aspect ratios ν^-1,


    B_2^∗ (ell) (D, ν^-1) = 12ν^-(D+1)   ∑_i=0^D [ [ D; i;   ] _2ℱ_1 (D+12,i2,D2;1-1ν^2) . 
           .  _2ℱ_1 (D+12,D-i2,D2;1-1ν^2) ]


is obtained. 
Using the Pfaff transformation <cit.> 



    _2ℱ_1 (a,b,c;z )    = (1-z )^-a _2ℱ_1 (a,c-b,c;zz-1) 
       =(1-z )^-b _2ℱ_1 (c-a,b,c;zz-1)



the relations 

    _2ℱ_1 (D+12,i2,D2;1-ν^2 ) = ν^-(D+1) _2ℱ_1 (D+12,D-i2,D2;1-1ν^2)


and

    _2ℱ_1 (D+12,D-i2,D2;1-ν^2 ) = ν^-(D+1) _2ℱ_1 (D+12,i2,D2;1-1ν^2)


result. 
Using Eqs. (<ref>) and (<ref>), Eq. (<ref>) can be written as 

    B_2^∗ (ell) (D, ν^-1) = 12ν^D+1   ∑_i=0^D [ [ D; i;   ] _2ℱ_1 (D+12,i2,D2;1-ν^2 ) . 
           .  _2ℱ_1 (D+12,D-i2,D2;1-ν^2 )]  ,

equal to Eq. (<ref>). 
Hence, the theorem 

    B_2^∗ (ell) (D, ν) = B_2^∗ (ell) (D, ν^-1)

results. This completes the proof. 





 §.§ Solids without inversion symmetry


Hyperspherical caps, hypercones, and truncated hypercones are convex, uniaxial solids of revolution without inversion symmetry for which
second virial coefficients B_2 and reduced second virial coefficients B_2^∗  can be obtained analytically from the geometric measures 
V_, S_, W_2, and R̃_ [Eq. (<ref>)].




  §.§.§ Hyperspherical caps


The second virial coefficients of hyperspherical caps can either be expressed in dependence on the critical angle θ_ or the aspect ratio ν. The latter quantities are related by 


    ν = {[ [1- cos( θ_2) ][2 sin(θ_2)]^-1                              :                        0 ≤θ_≤π;                               ;                12[1-cos(θ_2) ]                              :                      π≤θ_≤ 2 π ].
 

[Eqs. (<ref>), (<ref>), and  (<ref>)]. 




In Fig. <ref> (lhs), the second virial coefficients B_2 of hyperspherical caps are shown in dependence on the critical angle θ_. 
For critical angles θ_→ 0, a spherical hyperplate with radius r_ and second virial coefficient B_2 = (64/9 + 2/3π^2) r_ cap^4 results, while for θ_→ 2 π, a hypersphere with radius r_0 and second virial coefficient B_2 = 4 π^2 r_0^4 is obtained. 
For θ_ = π, simply r_ = r_0 = h with second virial coefficient B_2 = ( 43/24π^2 + 2 π + 16/9) r_0^4 results. 

The ratio of reduced second virial coefficients of hyperspherical caps and hyperlenses B_2^∗(ν)/B_2^∗ (hlen)(ν) is displayed in Fig. <ref> (rhs) in dependence on the aspect ratio ν.
The reduced second virial coefficients are nearly identical with a maximum ratio at ν≈ 0.59, a minimum ratio at ν≈ 0.22 and a second, less pronounced minimum ratio at ν≈ 0.95. 
In the limit ν→ 0, both geometries approach a spherical hyperplate. 









  §.§.§ Hypercones & truncated hypercones
 


The second virial coefficients B_2 and reduced second virial coefficients B_2^∗ of truncated hypercones are depicted in Fig. <ref> in dependence on the aspect ratio ν for selected ratios r_1/r_0. 
In the limit r_1/r_0 → 0, truncated hypercones approach hypercones. For the second virial coefficient B_2^(hcon) of hypercones, in the limit 

    lim_ν→∞B_2^(hcon)ν^2 r_0^4 = 149
 
the proportionality B_2 (ν≫1) ∝ν^2 is observed while for the reduced second virial coefficient in this limit


    lim_ν→∞B_2^∗ (hcon)ν = 143 π


the proportionality B_2^∗( ν≫ 1) ∝ν results. 




The second virial coefficients of both, hypercones and truncated hypercones, approach in the limit  

    lim_ν→ 0B_2r_0^4 = 649 + 23π^2


the expected value of a spherical hyperplate. 
In the limit r_1 / r_0 → 0, approaching hypercones with the reduced second virial coefficient B_2^∗ (hcon), in the limit 


    lim_ν→ 0[ ν B_2^∗ (hcon)] = 2 π + 643 π ,


the proportionality B_2^∗ (hcon) (ν≪ 1) ∝ν^-1 is observed for vanishing aspect ratios ν. 

With increasing aspect ratio ν, the second virial coefficient B_2 of truncated hypercones decreases with increasing ratio of radii r_1/r_0. 
The minima of reduced second virial coefficients B_2^∗ with respect to the aspect ratio ν decrease with rising ratio of radii r_1/r_0 and shift to larger aspect ratios.



 §.§ Excluded volume of D-dimensional spherocylinders


Using the general expression for quermassintegrals of D-dimensional, uniaxial spherocylinders in Euclidean spaces ℝ^D 

    W_i = [κ_D + 2 (ν-1)  D-iDκ_D-1] r_^D-i


[Eq. (<ref>)] and the Brunn-Minkowski theorem [Eq. (<ref>)], the excluded volume of identical particles can be written as


    V_^(scyl)   = 2^D V_ + 2^D-2(S_R̃_ - D V_)


for any dimension D, depending only on quermassintegrals of orders i=0, i=1 and i=D-1.


For identical, uniaxial D-dimensional spherocylinders with aspect ratio ν and radius r_, the rotation-averaged excluded volume V_^(scyl) can be written as


    V_^(scyl)   = (2 r_)^D [ κ_D + 2 (ν -1 ) κ_D-1 + (ν -1 )^2  D-1D κ_D-1^2κ_D] 
       = 2^D V_ + 2^D-2(S_R̃_ - D V_)


in Euclidean spaces ℝ^D with arbitrary dimension D.



Starting from the Brunn-Minkowski theorem


    V_(K) = 1κ_D∑_i=0^D [ D; i;   ]  W_i(K) W_D-i(K)


[Eq. (<ref>)] for the rotation-averaged excluded volume between any two identical convex solids K, the expression


    V_^(scyl) = 1κ_D   ∑_i=0^D [ D; i;   ]{[κ_D r_^D-i + 2 (ν-1 ) D-iDκ_D-1 r_^D-i] . 
           . [κ_D r_^i + 2 (ν -1 ) iDκ_D-1 r_^i ] }


results for D-dimensional spherocylinders using Eq. (<ref>). 
This can be rewritten as 


    V_^(scyl)   = ∑_i=0^D [ D; i;   ]κ_D r_^D  +∑_i=0^D [ D; i;   ] 2 ( ν -1 ) κ_D-1 r_^D  +  
         ∑_i=0^D [ D; i;   ] 4 ( ν -1 )^2 (D-i)iD^2κ_D-1^2κ_D r_^D   .


Using the binomial theorem 

    ∑_i=0^D [ D; i;   ]  =2^D  ,


for the excluded volume between identical, D-dimensional spherocylinders 


    V_^(scyl) =(2 r_)^D [κ_D + 2 (ν -1) κ_D-1] + (ν-1 )^2Dκ_D-1^2κ_D r_^D ∑_i=0^D [ D; i;   ] 4 (D-i) iD ,


is obtained.  With 

    ∑_i=0^D [ D; i;   ] 4 (D-i) iD = 2^D (D-1 )


immediately


    V_^(scyl) = (2 r_)^D [ κ_D + 2 (ν -1 ) κ_D-1 + (ν -1 )^2  D-1D κ_D-1^2κ_D]


results. Using the representation of geometric measures V_, S_, and R̃_ via quermassintegrals



    V_   = W_0 = κ_D r_^D + 2 (ν-1 ) κ_D-1 r_^D  , 
    
    S_   = D  W_1 = D κ_D r_^D-1 + 2 (ν -1 ) (D-1 ) κ_D-1 r_^D-1 ,
    R̃_   = 1κ_D W_D-1 = r_ + 2 (ν-1 ) 1Dκ_D-1κ_D r_ ,



the relation


    S_R̃_ - D V_ = 4 (ν-1 )^2 D-1Dκ_D-1^2κ_D r_^D


is obtained for D-dimensional spherocylinders [Eqs. (<ref>) and (<ref>)]. With
Eqs. (<ref>) and (<ref>), the excluded volume can be rewritten as

    V_^(scyl)   = 2^D V_ + 2^D-2(S_R̃_ - D V_)

[Eq. (<ref>)]. This completes the proof.  


As a unique feature of four-dimensional hyperspherocylinders, their excluded volume V_^(hscy) [Eq. (<ref>)] does not depend on the
particle volume V_. Using Eq. (<ref>), the second virial coefficient of D-dimensional, uniaxial spherocylinders reads as


    B_2^(scyl) = 2^D-1 V_ + 2^D-3(S_R̃_ - D V_)


and the reduced second virial coefficient B_2^∗ as 


    B_2^∗ (scyl) = 2^D-1 + 2^D-3(S_R̃_V_-D )


[Eq. (<ref>)]. 


The influence of the aspect ratio ν is visible from the reformulation 


    B_2^∗ (scyl) = 2^D-1[1+ D-1Dκ_D-1κ_D(ν-1 )^22 (ν-1 )+ κ_Dκ_D-1^-1]


with the limits 


    lim_ν→ 1 B_2^∗ (scyl)(D, ν) = 2^D-1


for a D-dimensional sphere [Eq. (<ref>)] and  


    lim_ν→∞B_2^∗ (scyl) (D,ν)ν = 2^D D-18 √(π)Γ(D/2 )Γ((D+1)/2 )


for an infinitely long D-dimensional spherocylinder. In the latter limit, the proportionality B_2^∗∝ν is obtained in accordance to the results in dimensions 2≤ D≤ 4
reported in the literature <cit.>. 
Since in the limit ν→∞ the contributions of the capping D-dimensional hemispheres are negligible (see Sec. <ref>), the limits 


    lim_ν→∞B_2^∗ (scyl) (D,ν)ν = lim_ν→∞B_2^∗ (cyl) (D,ν)ν


are identical for D-dimensional spherocylinders (scyl) and D-dimensional cylinders (cyl).  
In Table <ref> the results for these limits are provided for Euclidean spaces ℝ^D with D≤ 16. 




Instead of an infinitely long D-dimensional spherocylinder with radius r_, a rod with length l and radius r_=0 is an alternative description of a hard needle in ℝ^D <cit.>.  
The quermassintegrals W_i read as 


    W_i = {[        0        :  i ≤ D-2; κ_D-1D l        :  i = D-1;      κ_D        :      i=D ].


for such line segments <cit.>. 
With Eq. (<ref>), in ℝ^1 the excluded volume  V_ = 2 l and in ℝ^2 the rotation-averaged excluded volume V_=2l^2/π of hard rods are obtained. 
In Euclidean spaces ℝ^D with D≥ 3, the rotation-averaged excluded volume V_=0 of infinitely thin rods with length l vanishes <cit.>. 





§ SUMMARY AND OUTLOOK
 

In this work, analytical expressions for the geometric measures volume V_, surface area S_, second quermassintegrals W_2 and mean radius of curvature R̃_ are derived for selected convex solids of revolution in ℝ^4. 
The results are summarized in Tables <ref>, <ref>, and <ref>. 
Using Eqs. (<ref>), analytical expressions for so far unknown quermassintegrals W_i are obtained. 
Employing these quantities, with Eq. (<ref>) the intrinsic volumes υ_i of these convex solids 
of revolution are analytically accessible, too.

In addition to hyperspheres, hyperellipsoids of revolution, hyperspherocylinders, hypercylinders, and spherical hyperplates with already known quermassintegrals (Table <ref>), 
analytical expression for so far unknown quermassintegrals of hyperspindles, hyperlenses, hyperdoublecones, hypercones, truncated hypercones, and hyperspherical caps are
provided (Tables <ref> and <ref>). The latter geometries possess removable singularities in their surface curvature. While apical, zero-dimensional singularities do not contribute to
quermassintegrals, higher-dimensional singularities contribute as exemplarily shown in detail for hyperlenses. This contribution, in general depends on the critical angle enclosed within the ϵ-vicinity
of the respective singularity.  For uniaxial solids of revolution with infinitely large aspect ratios in ℝ^4, a general proportionality B_2^∗∝ν of reduced second virial coefficients to the aspect ratio arises whereas for infinitely thin oblate geometries, the general proportionality B_2^∗∝ν^-1 results. 

With known principal radii of curvature, the quermassintegrals are accessible for arbitrary convex solids in ℝ^4 [Eqs. (<ref>)]. The generalization to dimensions D>4 for geometries with continuous surfaces curvature [Eq. (<ref>)] is straight forward. Possible contributions of singularities can
be determined analogously as demonstrated in ℝ^3 <cit.> and ℝ^4 (this work).


For selected geometries, general expressions for their quermassintegrals are compiled in Table <ref>. Using such general expressions, the parity B_2^∗(ν)=B_2^∗(ν^-1) for uniaxial D-dimensional ellipsoids is proven. Additionally, 
for D-dimensional, uniaxial spherocylinders the dependence of the reduced second virial coefficient B_2^∗ on at most three quermassintegrals W_0, W_1, and W_D-1 is proven, too. 

A remaining task is the derivation of a general expression for quermassintegrals of D-dimensional doublecones in ℝ^D needed to prove or disprove the conjecture that in 
even dimensions D reduced second virial coefficients of cylinders and doublecones are identical at aspect ratio ν=1.
























 






























50
#1ISBN #1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1<https://doi.org/#1>et al.#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1<><#>1#1#1#1#1#1#1#1#1#1#1#1#1#1PreBibitemsHook


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