Metric and geometric relaxations of self-contracted curves
Aris Daniilidis Robert Deville Estibalitz Durand Cartagena




Metric and geometric relaxations of self-contracted curves 




A. Daniilidis, R. Deville, E. Durand-Cartagena




Abstract Self-contractedness (or self-expandedness,
depending on the orientation) is hereby extended in two natural ways giving
rise, for any λ∈-1,1), to the metric notion of λ-curve and the (weaker) geometric notion of λ-cone property
(λ-eel). In the Euclidean space ℝ^d it is established
that for λ∈-1,1/d) bounded λ-curves have finite
length. For λ≥ 1/√(5) it is always possible to construct bounded
curves of infinite length in ℝ^3 which do satisfy the λ-cone property. This can never happen in ℝ^2 though: it is
shown that all bounded planar curves with the λ-cone property have
finite length.




Key words Self-contracted curve, self-expanded curve,
rectifiability, length, λ-curve, λ-cone property.




AMS Subject Classification  Primary 28A75, 52A38 ;
Secondary 37N40, 53A04, 53B20, 52A41.










§ INTRODUCTION


Self-contracted curves have been introduced in <cit.>. They attract a lot
of interest, since they are intimately linked to convex foliations
(<cit.>, <cit.>, <cit.>), to the proximal algorithm of a convex
function and the gradient flow of a quasiconvex potential in a Euclidean space
(<cit.>, <cit.>) and recently to generalized flows in CAT(0) spaces
(<cit.>). The main feature of this notion is its simple purely metric definition, which
inspires developments in more general settings:


Let (M,d) be a metric space and I⊂ℝ be an interval. A curve
γ:I→ M is called self-contracted, if for all τ∈ I, the
map t↦ d(γ(t),γ(τ)) is non-increasing on I∩
(-∞,τ].


The length of a curve γ is defined as

    ℓ(γ):=sup{∑_i=0^m-1d(γ(t_i),γ(t_i+1
    ))},

where the supremum is taken over all finite increasing sequences t_0<t_1<⋯<t_m lying in I. 
The curve γ is called rectifiable,
if its total variation is locally bounded around any t∈ I, that is, its
length is locally finite. 

Rectifiability and asymptotic behaviour are central questions in the study of
self-contracted curves. It is shown in <cit.> that self-contracted curves
(are rectifiable and) have finite length whenever M is a bounded subset of
the 2-dimensional Euclidean space. Based on ideas of <cit.>, the
aforementionned result was extended in <cit.>, and independently in
<cit.>, to any finite dimensional Euclidean space. In <cit.> a
further extension has been established encompassing the case where M is a
compact subset of a Riemannian manifold. In <cit.> the result of <cit.>
has been generalized for 2-dimensional spaces equipped with other (smooth)
norms. This has been the first result of this type outside a
Euclidean/Riemannian setting. An important breakthrough is eventually achieved
in <cit.> by establishing (rectifiability and) finite length for all
self-contracted curves contained on a bounded subset of any finite dimensional
normed space. Finally, rectifiability of self-contracted curves in Hadamard
manifolds and CAT(0) spaces is established in <cit.>.



The aforementioned results remain valid if we replace the assumption
“γ self-contracted” by the assumption
“γ self-expanded”. A curve γ is
called self-expanded if for all τ∈ I, the map t↦
d(γ(t),γ(τ)) is non decreasing on I∩τ,+∞),
or equivalently, when the curve γ:-I→ M given by
γ(t)=γ(-t) is self-contracted. Thus,
γ:I→ℝ^d is self-expanded 
if for every t_1≤ t_2≤ t_3 in I we have

    d(γ(t_1),γ(t_2))≤ d(γ(t_1),γ(t_3)).

In the Euclidean
setting, there is a nice geometric interpretation of self-expandedness (see
<cit.>). A differentiable curve is self-expanded if and only
if

    ⟨γ^'(t),γ(u)-γ(t)⟩≤0 
    for all u∈ I such that u < t,

which geometrically means that the tail of the curve (the past) is always
contained in half-space (cone of aperture π).
The notion of self-expandedness therefore admits the following two natural generalizations.
Let us fix -1≤λ<1. 
A curve γ:I→ℝ^d is called λ-curve  
if for every t_1≤ t_2≤ t_3 in I we have

    d(γ(t_1),γ(t_2))≤ d(γ(t_1),γ(t_3))+λ
    d(γ(t_2),γ(t_3)).

If γ is continuous and admits right derivative at each point,
we say that γ has the λ-cone-property if,
for every t<τ in I, we have, denoting γ'(τ) the right derivative,

    ⟨γ'(τ),γ(t)-γ(τ)⟩≤λ ||γ'(τ)|| ||γ(t)-γ(τ)|| .

As a matter of the fact, the λ-cone property will be defined more generally, for merely continuous curves 
using (forward) secants, see Definition <ref> and it will be shown that every λ-curve has the λ-cone property (c.f. Proposition <ref>). However there exist smooth curves satisfying the
latter property for some λ_0<1 without being λ-curves for
any λ∈-1,1) (c.f. Example <ref>).
In this work we establish the following results: 



  * if |·| is an equivalent norm to the Euclidean norm ||·||,
then there exists λ∈0,1) such that every |·
|-self-expanded curve is a ||·||-λ-curve
(Proposition <ref>); 


  * for λ<1/d every bounded λ-curve (is rectifiable and) has
finite length (Theorem <ref>);


  * for λ≥1/√(5) there exists a bounded curve in ℝ^3 with infinite length satisfying the λ-cone property
(Theorem <ref>).

Nonetheless due to topological obstructions for d=2 we have:



  * for any λ<1, bounded planar curves with the λ-cone
property (and a fortiori λ-curves) have finite length
(Theorem <ref>). 

Combining the first and the last statement, we readily obtain that all bounded
planar self-contracted curves (under any norm) are rectifiable and have finite
length. This clearly generalizes the result of <cit.>, but it is contained
in the result of <cit.> that asserts that the same holds in any dimension.
Notice that the asymptotic behaviour of both λ-curves and curves with
the λ-cone property remains unknown in ℝ^d for d≥3
and λ∈ [1/d,1/√(5)).



Notation. Let us fix our notation. Throughout this work ℝ^d 
will denote the d-dimensional Euclidean space endowed with the
Euclidean norm ||·|| and the scalar product ⟨·,·⟩.
We denote by 𝕊^d-1 the unit sphere of ℝ^d, and by
B(x,r) (respectively, B(x,r)) the open (respectively, closed)
ball of radius r>0 and center x∈ℝ^d. A (convex) subset C
of ℝ^d is called a (convex) cone, if for every x∈ C and r>0
it holds rx∈ C. If A is a nonempty subset of ℝ^d, we
denote by int(A) its interior, by conv (A) its convex
hull and by diam A:=sup {d(x,y):x,y∈ A} its diameter.

Given a closed convex subset K of ℝ^d, the normal cone
N_K(u_0) of K at u_0∈ K is the following closed convex cone (see
<cit.> e.g.):

    N_K(u_0)={v∈ℝ^n:⟨ v,u-u_0⟩≤0,∀ u∈
    K}.

Notice that u_0∈ K is the projection onto K of all elements of the
form u_0+tv, where t≥0 and v∈ N_K(u_0). In the particular
case that K is a closed convex pointed cone (that is, K contains no
lines), then its polar (or dual) cone 

    K^o:=N_K(0)={v∈ℝ^n:⟨ v,u⟩≤0,∀ u∈
    K}

has nonempty interior and the bipolar theorem holds: K^oo=K. For δ>0 sufficiently small,
we denote by K_δ the δ-enlargement of the cone K, that is,
the closed convex cone generated by the set (  K∩𝕊^d-1)
+B_δ, where B_δ:=B(0,δ). Notice that

    ((K_δ)^o∩𝕊^d-1)  +B_δ  ⊂ K^o.





We define the aperture A(S) of a nonempty subset S⊂𝕊
^d-1 by

    A(S):=inf{ ⟨ u_1,u_2⟩ : u_1,u_2∈
    S }  .

Based on the above notion, we define the aperture 𝒜(C) of a
nontrivial convex pointed cone C as follows:

    𝒜(C)=arccos (  A(C∩𝕊^d-1))  .

Given v∈𝕊^d-1 and α∈0,π), we define the
“open" cone directed by v as follows:

    C(v,α)={  u∈ℝ^d:  ⟨ u,v⟩> ||u|| cosα}∪{0}.

Notice that if α<π/2, the above cone is convex and has aperture 2α. Given x∈ℝ^d, we adopt the notation

    C_x(v,α):=x+C(v,α).





A mapping γ:I=[0,T_∞)→ℝ^d, where
T_∞∈ℝ∪{+∞} is referred in the sequel as a
curve. Although the usual definition of a curve comes along with continuity
and injectivity requirements for the map γ, we do not make these prior
assumptions here. By the term continuous (respectively, absolutely
continuous, Lipschitz, smooth) curve we shall refer to the corresponding
properties of the mapping γ:I→ℝ^d. A curve
γ is said to be bounded if its image, denoted by Γ=γ(I),
is a bounded set of ℝ^d.



For t∈ I we denote by Γ(t):={γ(t^')∈Γ:t^'≤ t} the initial part of the curve and by

    K(t)=cone (Γ(t)-γ(t))

the closed convex cone generated by Γ(t). In particular

    Γ(t)⊂γ(t)+K(t)

Notice further that K(t) contains the set sec^-(t) of (all
possible limits of) backward secants at γ(τ) which is defined as
follows (see <cit.>):

    sec^-(t):={  q∈𝕊^d-1:q=lim_t_k↗
    t^- γ(t_k)-γ(t)/||γ(t_k)-γ(t)||}
    ,

where the notation {t_k}_k↗ t^- indicates that {t_k}_k→ t and t_k<t for all k. 

The set sec^+(t) of all possible limits of forward secants at
γ(t) is defined analogously:

    sec^+(t):={  q∈𝕊^d-1:q=lim_t_k↘
    t^+ γ(t_k)-γ(t)/||γ(t_k)-γ(t)||}
    ,

where the notation {t_k}_k↘ t^+ indicates that {t_k}_k→ t and t<t_k for all k. Compactness of 𝕊^d-1 guarantees that both sec^-(t) and sec^+(t)
are nonempty. If γ:I→ℝ^d is differentiable at
t∈ I and γ^'(t)≠0, then sec^+(t)={γ^'(t)/||γ^'(t)||}.

In this work we introduce two new notions, depending on a parameter
λ∈-1,1). For each value of λ we obtain
the class of λ-curves and the class of curves with the λ-cone
property. We associate to these classes an angle α∈(0,π] via the
relation

    α=arccos(λ).

As we shall see, the above classes enjoy interesting geometric properties
which can be described in terms of the angle α. (For λ=0, which
corresponds to the angle α=π/2, the above classes coincide and yield
the class of self-expanded curves.)







§ Λ-CURVES AND CURVES WITH THE Λ-CONE PROPERTY


 A curve γ:I→ℝ^d is called λ-curve (-1≤λ<1) if for every t_1≤
t_2≤ t_3 in I we have

    d(γ(t_1),γ(t_2))≤ d(γ(t_1),γ(t_3))+λ
    d(γ(t_2),γ(t_3)).




The above definition yields that every λ-curve is necessarily injective and
cannot admit more than one accumulation point. Based on this, one can easily
see that every λ-curve has at most countable discontinuities. Setting
λ=1 to (<ref>) yields the triangle inequality of the
distance (hence no restriction) while λ=-1 corresponds to segments. On
the other hand, for λ=0 we recover the definition of a self-expanded
curve. The following result shows that the study of
self-contracted/self-expanded curves with respect to a non-Euclidean norm can
be shifted to the study of λ-curves in the Euclidean setting.

Let · be
an Euclidean norm in ℝ^d and |·| be another norm in ℝ^d. 
Then there exists λ<1
(depending on the equivalence constant of the norms) such that every |·
|-self-expanded curve is a ·-λ-curve.


Proof. Since the norms |·| and ‖·‖ are
equivalent and since the properties of being self-expanded or being a
λ-curve are invariant by homothetic transformation, we may assume that
there exists δ>0 such that for all x∈ℝ^d, δ‖
x‖≤|x|≤‖ x‖. Let t_0<t_1<t_2 in I and set
x_0=γ(t_0), x=γ(t_1) and y=γ(t_2). It follows by
assumption that

    |x-x_0|≤|y-x_0|.

To establish the result it is sufficient to prove that there exists
λ<1 such that for all choices of x_0,x,y satisfying (<ref>),
we have

    ‖ x-x_0‖≤‖ y-x_0‖+λ‖ x-y‖.

By translation, we may, and do assume, that x_0=0. Moreover, by
homogeneity, we can assume |y|=1. Set

    B={z∈ℝ^d; |z|≤δ}  and  C_y
    ={y+t(y-z); ‖ z‖<δ, t>0}.



We claim that B∩ C_y=∅. Indeed, fix z such that ‖
z‖<δ. The function φ:ℝ→ℝ defined by
φ(t)=| y+t(y-z)| is convex, φ(-1)<1 and φ
(0)=1, hence φ(t)>1 whenever t>0, that is, y+t(y-z)∉ B.
Since this is true for all z satisfying ‖ z‖<δ, the claim is
proved. 

Therefore Proposition <ref> is a consequence of the following lemma.


There exists λ<1 such that, whenever 1≤‖ y‖≤1/δ,
‖ x‖≤1/δ and x∉ C_y, then ‖ x‖-‖
y‖≤λ‖ x-y‖.


Proof. Set u=x-y and Γ_y
:={t(y-z); ||z||<δ, t>0}. We claim that there exists ρ<1 such
that, whenever 1≤||y||≤1/δ and u∈ℝ^dΓ_y, then

    ⟨ u,y⟩≤ρ ||u||·||y||.



Indeed, since ℝ^dΓ_y is a cone, it is enough to
establish (<ref>) when ‖ u‖=‖ y‖. Let us denote by
c(u,y) the cosine of the angle of the two vectors u and y. The condition
u∉Γ_y yields ‖ u-y‖≥δ. Then we obtain ‖
u-y‖^2=‖ y‖^2(2-2c(u,y))≥1, which yields

    c(u,y) ≤ 1-δ^2/2‖ y‖^2 ≤ 1-δ
    ^4/2:=ρ < 1.

This proves the claim. 

Since ||y+u||^2⩽||y||^2+||u||^2+2||y|| ||u|| ρ, we deduce
from (<ref>) that

    ||x||-||y||=||y+u||-||y|| ≤ ||y||(  √(1+2ρ||u||/||y||+||u||^2/||y||^2)-1)  .

Since ||u||=||x-y||≤||x||+||y||≤2/δ and ||y||≥|y|=1, we have
t=||u||/||y||∈0,2/δ]. Notice that taking λ<1
sufficiently close to 1, we ensure that for all t∈0,2/δ] it
holds

    √(1+2ρ t+t^2)-1≤λ t.

Therefore we conclude that

    ||x||-||y||≤λ||u||=λ||x-y||.

The proof is complete. □



From now on we consider exclusively a Euclidean setting. An important feature
of the notion of λ-curve is the following property:

Let γ:I→ℝ^d be a λ-curve. Then, γ is λ-uniformly
non-collinear, that is, for every s,u,t∈ I such that s,u≤ t we have

    ⟨γ(u)-γ(t)‖γ(u)-γ(t)‖
    ,γ(s)-γ(t)‖γ(s)-γ(t)‖⟩ > -λ  (   > -1 )   .




Proof. Assume that u<s<t. Because γ is λ-curve we have that

    d(γ(u),γ(s)) ≤ d(γ(u),γ(t)) + λ d(γ(s),γ(t))

Consider the triangle of vertices γ(t), γ(u) and γ(s) and
set c=d(γ(u),γ(s)), a=d(γ(u),γ(t)) and b=d(γ
(s),γ(t)). The previous equation now reads c≤ a+λ b, and
after squaring both sides we get

    c^2≤ a^2+λ^2b^2+2λ ab.

Evoking the law of cosine c^2=a^2+b^2-2abcosφ we deduce

    cosφ=a^2+b^2-c^2/2ab (<ref>)≥ (1-λ^2)b^2-2λ ab/2ab > -λ,

that is, the angle φ between the vectors

    γ(u)-γ(t)‖γ(u)-γ(t)‖  and  γ(s)-γ(t)‖γ(s)-γ(t)‖

is strictly less than π-α (α=arccos(λ) is given by
(<ref>)).□



Before we proceed, we give the following definition.

 Let λ∈-1,1)
and α=arccos(λ). We say that a continuous curve γ
:I→ℝ^d satisfies the λ-cone property if for
every t∈ I and for every q_t^+∈sec^+(t) it holds

    ⟨ q_t^+,γ(u)-γ(t)/‖γ(u)-γ
    (t)‖⟩≤λ,  for all u<t.




In other words, recalling (<ref>), the set Γ(t)-γ(t)
does not intersect the cone C(  q_t^+,α) directed by
q_t^+ and of aperture 2α expect at 0, that is, for every t∈
I

    (  γ(t) +⋃_q_t^+∈sec^+(t)C(
    q_t^+,α)  )    ⋂  Γ(t) = {γ(t)}  .




We shall now consider a second important feature of the class of (continuous)
λ-curves.


Every continuous λ-curve has the λ-cone property.


Proof. Fix t∈ I, let u<t, q_t^+∈sec
^+(t) and choose {t_k}_k↘ t such that

    γ(t_k)-γ(t)/‖γ(t_k)-γ(t)‖ ⟶ q_t^+.

Since γ is a λ-curve we have

    ‖γ(t)-γ(u)‖≤‖γ(t_k)-γ(u)‖
    +λ‖γ(t_k)-γ(t)‖,

yielding

    ‖γ(t_k)-γ(u)‖-‖γ(t)-γ(u)‖/||γ(t_k)-γ(t)|| ≥ -λ

Set Φ(X)=||X||, X_k=γ(t_k)-γ(u) and X=γ
(t)-γ(u). Then the above inequality reads

    Φ(X_k)-Φ(X)/||X_k-X|| ≥ -λ.

Since the norm is differentiable around the segment [X,X_k]:={tX+(1-t)X_k: t∈ [0,1]}, applying the
Mean Value theorem we obtain θ_k∈0,1) such that

    Φ(X_k)-Φ(X)=DΦ(X+θ_k(X_k-X))(X_k-X)=⟨X+θ_k(X_k-X)/‖ X+θ_k(X_k-X)‖,X_k
    -X⟩ .

Combining the above formulas and taking the limit as k→∞ we
get

    ⟨γ(t)-γ(u)/‖γ(t)-γ(u)‖
    , q_t^+⟩ ⩾ -λ.

The above is equivalent to (<ref>) and the proof is complete.□



The following example reveals that there exist C^1curves satisfying the
λ-cone property but failing to satisfy the non-collinearity property.
Therefore these curves cannot be λ-curves for any value of the
parameter λ∈-1,1).


 Let γ:[-3π/2,1+π]→ℝ^3 be
defined by

    γ(t)  ={[                    (0,-sin t,-cos t),                     if t∈-3π/2,-π/2],;           (-1/2(1+cos2t),1,1/2sin2t),                        if t∈
    -π/2,0],;                             (-1,1,t),                            if t∈0,1],; (-1,1/2(1+cos2(t-1)),1+1/2sin2(t-1)),                       if 
    t∈1,1+π/2],;             (-sin(t-1),0,1+cos(t-1)),                      if t∈1+π/2,1+π]. ].

It is easy to check that γ is C^1-smooth. Moreover, γ fails
to satisfy the non-collinearity property: indeed, γ(1+π)=(0,0,0) is
the midpoint of the segment [γ(-3π/2),γ(-π/2)]. Hence, by
Proposition <ref>, γ cannot be a λ-curve for any
value of the parameter λ<1. On the other hand, any tangent line

    {γ(t)+sγ^'(t); s∈ℝ}

meets the curve {γ(τ); τ∈[-3π/2,1+π]} only at the point
γ(t). Therefore, by a simple compactness argument, there exists
λ_0<1 for which γ satisfies the λ_0-cone property.







§ LENGTH OF Λ-CURVES


Before we proceed we recall from <cit.> the following result (we provide
a proof for completeness).


 Let Σ⊂𝕊^d-1 (the unit sphere of
ℝ^d, d>1) and assume that for λ<1/d it holds

    ⟨ x,x^'⟩≥-λ,  for all x,x^'∈Σ.

Then Σ is contained in a half-sphere (therefore it generates a closed
convex pointed cone).


Proof. Notice that the conclusion holds if and only if
0∉conv (Σ). Let us assume that 0∈conv 
(Σ). Then by Caratheodory theorem, there exist α_0,α
_1,⋯,α_d≥0 and x_0,x_1,⋯,x_d∈𝕊^d-1
such that

    ∑_i=0^dα_i=1  and  ∑_i=0^dα_i
    x_i=0.

It follows that for j∈{0,1,…,d},

    0=⟨0,x_j⟩=∑_i=0^dα_i⟨ x_i
    ,x_j⟩≥α_j-λ∑_i≠ jα_i=α_j
    -λ(1-α_j).

Summing up for all j∈{0,1,…,d} we get 0≥1-λ(d+1-1),
which contradicts the assumption λ<1/d.□



Recalling the notation of (<ref>), (<ref>) and (<ref>), and
assuming λ<1/d we obtain the following result (as a straightforward
combination of Lemma <ref> with Proposition <ref>).

Let -1≤λ<1/d and α=arccos(λ). Then for every t∈ I, the
initial part Γ(t) of a λ-curve γ is contained in a
closed convex cone K(t) of aperture at most π-α centered at γ(t).
In other words,

    Γ(t)⊂γ(t)+K(t)  and  𝒜(K(t))≤π-α.





To sum up, given a continuous λ-curve γ,
Proposition <ref> ensures that its initial part Γ(t)
avoids the union of all cones centered at γ(t) and directed by forward
secants of γ at t, see (<ref>), while
Corollary <ref> asserts that, provided λ<1/d, the initial
part of the curve Γ(t) is itself contained in the closed convex pointed
cone γ(t)+K(t), centered at γ(t). The following proposition
asserts that an even stronger property is satisfied.

 Let γ:I→ℝ^d be a continuous λ-curve, with λ∈-1,1/d) and α=arccos(λ). Then it holds:

    (  ⋃_q_t^+∈sec^+(t)C(  q_t^+,α)  )    ⋂   K(t)  =  {0},     for all  t∈ I.




Proof. Assume towards a contradiction that for some
q_t^+∈^+(t) there exists q∈ C(  q_t^+,α)  ∩ K(t), q≠0. 
This yields, in view of Proposition <ref>, that int K(t) is nonempty. Therefore, since q satisfies the open
condition

    ⟨ q_t^+,q⟩ > λ=cosα,

there is no loss of generality to assume that q∈int K(t).
Therefore, there exist t_1<t_2<…<t_d<t and {μ_i}_i=1
^d⊂ℝ_+ such that

    u_i:=γ(t_i)-γ(t)/‖γ(t_i)-γ(t)‖  and   q=∑_i=1^dμ_i u_i.

Fix ε>0 such that ⟨ q_t^+,q⟩>λ
+3ε. By continuity, there exists δ>0 such that for all
s∈(t,t+δ) the vectors

    ũ_i:=γ(t_i)-γ(s)/‖γ(t_i)-γ
    (s)‖,   i∈{1,…,d},

are sufficiently close to {u_i}_i=1^d to ensure that

    ⟨ q_t^+,q̃⟩>λ+2ε,    where   q̃=∑_i=1^nμ_i ũ_i.

Take now s∈(t,t+δ) in a way that the vector q̂=(
‖γ(s)-γ(t)‖)  ^-1(  γ(s)-γ(t)) is sufficiently close to the secant q_t^+ so that ⟨q̂,q̃⟩>λ+ε or equivalently, ⟨-q̂,q̃⟩<-λ-ε. Since q̃,-q̂∈
K(s)∩𝕊^d-1, we deduce that 𝒜(K(s))>π-α, which
contradicts Corollary <ref> for s=t. □



We shall finally need the following lemma.


 Let γ:I→ℝ^d be a
continuous λ-curve, with λ∈-1,1/d) and α
=arccos(λ). Then there exists ρ>0 such that for every t∈ I
and q_t^+∈sec^+(t), there exists ξ_t∈𝕊
^d-1 satisfying

    ⟨ξ_t,u⟩≤-ρ<0,  for all u∈ K(t)

and

    ⟨ξ_t,q_t^+⟩ ≥ ρ >0 .




Proof. Let δ≤√(2(1-λ)) and ρ
=δ/2. Then for every t∈ I and q_t^+∈sec^+(t), we
have 𝕊^d-1∩ B(q_t^+,δ)⊂ C(q_t^+,α). We deduce
from Proposition <ref> that the δ-enlargement of the cone
K(t) satisfies:

    K(t)_δ∩sec^+(t)=∅.

Setting Ñ(t)=(K(t)_δ)^o and  N(t)=K(t)^o (the polar of
K(t)_δ and K(t) respectively), we deduce by (<ref>) that

    B̅(ξ,δ)∩𝕊^d-1⊂ N(t),   for every ξ∈Ñ(t)∩𝕊^d-1.

Let us now fix q_t^+∈sec^+(t). Then by the bipolar theorem
we get q_t^+∉Ñ(t)^o=K(t)_δ, that is, there exists
ξ̃∈Ñ(t)∩𝕊^d-1 such that ⟨ξ̃,q_t
^+⟩>0. Maximizing the functional q_t^+ over the closed ball B̅(ξ̃,ρ) 
we obtain ξ_t∈𝕊^d-1 such that (<ref>) holds. Since B(ξ
_t,ρ)⊂ B(ξ̃,δ)⊂ N(t), we easily deduce that
(<ref>) also holds. □







We are now ready to prove the main result of this section.

Every continuous λ-curve
γ:I→ℝ^d with λ<1/d is rectifiable. In
particular, bounded λ-curves with λ<1/d have finite length.


Proof. We may assume that I=[0,+∞) and that
γis bounded. Set η=ρ/3, where ρ is given by Lemma
<ref>. Since 𝕊^d-1 is compact, there exists an
η-net ℱ:={ξ_1,⋯,ξ_N}, satisfying that for
every v∈𝕊^d-1, there exists i∈{1,⋯,N} such that
⟨ v,ξ_i⟩>η (that is, v is η-close to some ξ
_i∈ℱ). Then we deduce from Lemma <ref> that for
every t∈ I and q_t^+∈^+(t), there exists ξ_i∈ℱ such that

    ⟨ξ_i,q^+⟩>2η    and    ⟨ξ_i
    ,u⟩≤-2η<0,  for all u∈ K(t).

Reasoning by contradiction we can prove the existence of some δ_t>0
such that for every s∈ t,t+δ_t) there exists q_t,s^+∈sec^+(t) such that

    ‖γ(s)-γ(t)/||γ(s)-γ(t)||-q_t,s
    ^+‖ <η.

Combining the above we deduce that for every t∈ I and s∈
t,t+δ_t), there exists ξ_i∈ℱ such that

    ⟨ξ_i,γ(s)-γ(t)⟩≥ η ‖γ
    (s)-γ(t)‖.

On the other hand, it follows directly from (<ref>) that for every
τ∈0,t)

    ⟨ξ_i,γ(t)-γ(τ)⟩≥ η ‖γ
    (t)-γ(τ)‖.

Considering for i∈{1,…,N} the projection operator

    {[        π_i:ℝ^d→ℝξ_i; π_i(x)= ⟨ξ_i,x⟩ ξ_i ].

we define W_i(t) to be the width of the projection of the initial part of
the curve Γ(t) onto ℝξ_i, that is,

    W_i(t):=ℋ^1(π_i(Γ(t))),   t∈ I,

where ℋ^1 denotes the 1-dimensional Lebesgue measure. Notice
that ℋ^1(π_i(Γ(t))) is simply the length of the bounded
interval π_i(Γ(t)) of ℝξ_i. It follows readily that
for every i∈{1,…,N} the function t↦ W_i(τ) is
non-decreasing on [0,T_∞) and bounded above by r:=diam
(γ(I)). Therefore, the function

    W_ℱ(t):=∑_i=1^NW_i(t),

is non-decreasing on I and bounded above by Nr. We now deduce from
(<ref>) and (<ref>) that for every t∈ I there exists δ
_t>0 such that for all s∈ t,t+δ_t) we have

    W_ℱ(s)-W_ℱ(t)≥η ‖γ(s)-γ
    (t)‖.

The result follows via a standard argument if we establish that for any
a,b∈ I with a<b it holds:

    W_ℱ(b)-W_ℱ(a)≥η ‖γ(b)-γ
    (a)‖.

Let us assume, towards a contradiction, that (<ref>) does not hold, that
is,

    W_ℱ(b)-W_ℱ(a)+ε<η ‖γ
    (b)-γ(a)‖,   for some ε>0.

Set σ(t)=sup{s>t: (<ref>)  holds}, for t∈ a,b). Then
our assumption yields that for every t∈ a,b) we have a≤
t+δ_t≤σ(t)<b. Using transfinite induction we construct a
(necessarily) countable set Λ={t_μ}_μ≤ς̂ by
setting t_1=a, t_μ=σ(t_μ^-) if μ=μ^-+1 is a
successor ordinal, and t_μ=sup{t_ν:ν<μ} if μ is a limit
ordinal and we stop when t_ς̂=b. Let now {ε
_n}_n∈ℕ⊂(0,ε) with 
∑_n∈ Nε_n=εη^-1. Let i:Λ→ℕ be
an injection of Λ into ℕ. Then denoting by μ^+ the
successor of μ, we obtain by continuity, that for each ordinal μ there
exists t_μ≤ s_μ<σ(t_μ):=t_μ^+ such that
||γ(s_μ)-γ(t_μ^+)‖<ε_i(μ). We deduce
by (<ref>):

    ‖γ(b)-γ(a)‖   ≤∑_μ∈Λ‖γ(t_μ^+)-γ(t_μ)‖≤∑_μ∈Λ(  ‖γ(s_μ)-γ(t_μ)‖+ε_i(μ))
    
       ≤1/η(
    ∑_μ∈Λ(  W_ℱ(s_μ)-W_ℱ(t_μ))
     + ε)  ≤1/η(  W_ℱ(b)-W_ℱ(a)+ε)  ,

which contradicts (<ref>).□



 The above proof reveals that the length ℓ(γ) of any
λ-curve lying in a set of diameter r is bounded by the
quantity N·η^-1· r. Since the constant η>0 is
determined in Lemma <ref>, it only
depends on λ and the dimension d of the space (in
particular, it is independent of the specific λ-curve γ). Since
N (the cardinality of the net ℱ) also depends exclusively on η and the
dimension d, we conclude that for a given λ∈-1,1/d) there exists
a prior bound for the lengths of all λ-curves γ lying inside a
prescribed bounded subset of ℝ^d.



A close inspection of Theorem <ref> shows that the proof depends exclusively on (<ref>)–(<ref>) which in turn depend on (<ref>). Therefore, every bounded continuous curve γ satisfying (<ref>) has finite length.
 










§ A BOUNDED CURVE WITH THE Λ-CONE PROPERTY AND INFINITE LENGTH


In this section we consider continuous right differentiable curves
γ:I→ℝ^d satisfying the λ-cone property
(Definition <ref>). In the sequel we denote by γ
^'(τ) the right derivative of γ at the point τ and we
assume this derivative is nonzero. Observe that in this case we have

    ^+(t)={γ'(t)/‖γ'(t)‖}  .

So γ satisfies the λ-cone property
if,  for all t,τ∈ I with t<τ, (<ref>) holds, or equivalently:

    ⟨γ'(τ),γ(t)-γ(τ)⟩≤λ ||γ'(τ)|| ||γ(t)-γ(τ)|| .

This means that the angle between the vectors γ'(τ) and 
γ(t)-γ(τ) is greater or equal to α, where α=arccos(λ).

We simplify the notation by setting

    C(t,α):=γ(t)+C(  γ'(t)/‖γ'(t)‖,α)  .

A curve γ satisfying the above property will be also called a λ-eel. The reason is as follows: the set Γ(τ):={γ
(t); t∈ I, t<τ} is the apparent body (or tail) of a λ-eel at
time τ going out of a hole. The cone C(τ,α) represents what the
λ-eel can see at time τ. The λ-cone property just says
that the λ-eel never sees its apparent tail. Notice that π/2-eels
correspond to self-expanded curves. Therefore, if the range of γ is
bounded and γ is a π/2-eel, then its length is finite (<cit.>,
<cit.>).




Recall from the introduction that a curve γ is
self-expanded if for all τ∈ I, the map t↦
d(γ(t),γ(τ)) is non decreasing on I∩τ,+∞). The following lemma illustrates that one can also associate a Lyapunov function to  λ-eels.


If γ:I→ℝ^d is a λ-eel, then the function 

    t↦γ(t_1)-γ(t)+λℓ(γ_|[t_1,t])

is non-decreasing on I∩[t_1,∞).



By definition,

    d/dτ(γ(τ)-γ(t))=⟨γ'(τ),γ(τ)-γ(t)/γ(τ)-γ(t)⟩≥ -λγ'(τ)  ∀  t<τ.
 
For t<t_1<t_2, integrating for τ∈ [t_2, t_3] we obtain

    ∫_t_2^t_3d/dτ(γ(τ)-γ(t))dτ≥ -λ∫_t_2^t_3γ'(s)ds  ∀  t<τ,
 
which implies

    γ(t_3)-γ(t)-γ(t_2)-γ(t)≥ -λℓ(γ_|[t_2,t_3]).
 
Since ℓ(γ_|[t_2,t_3])= ℓ(γ_|[t_1,t_3])- ℓ(γ_|[t_1,t_2]) the conclusion follows.





Our main aim now is to prove the following result.

 Assume 
λ=1/√(5) 
(i.e. α=arccos1/√(5)), and let B=B(0,1)
the unit ball of ℝ^3. Then, there
exists a λ-eel γ:[0,+∞)→ B of infinite length.
Moreover lim_t→∞γ(t) exists.


The proof of Theorem <ref> is constructive: the construction will be
carried out in three steps organized in subsections. Let us mention that the
result remains true if we require γ to be 𝒞^1-smooth (and
probably even 𝒞^∞-smooth), but the construction would then
become less transparent. Before we proceed, let us make the following remark.


Let us denote by λ_∗ the infimum of all λ for which there
exists a bounded λ-eel of infinite length inside the unit ball of
ℝ^3. Since for λ=0 we obtain a self-expanded curve, it
follows from the above theorem that 0≤λ_∗≤1/√(5). Notice that we
cannot readily conclude that λ_∗ is strictly greater than 0. (Nonetheless, according to <cit.> or <cit.>, for λ=0 bounded λ-eels have finite length.)
 






 §.§ Helicoidal maps



Let us start by constructing a helicoidal curve along the z-axis,
which is self-expanded.


 There exists a positive constant μ<1/2 such that, if
γ:ℝ→ℝ^3 is a spiral of the form

    γ(t)=(  r cos t, r sin t, μ rt)  ,   t∈ℝ,

then γ is self-expanded (hence γ satisfies the λ-cone
property for all λ∈0,1)).


Proof. Let γ:ℝ→ℝ^3
be a spiral along a cylinder of radius r>0 of the form (<ref>) and let
us show that γ is a self-expanded curve. By symmetry, this amounts to verify
that

    a(t):=⟨γ'(0),γ(t)-γ(0)⟩≤0,  for
    all  t<0.

We check easily that γ(0)=(  r,0,0) and γ̇(0)=(  0,r,μ r), so that a(t)=r^2(sin t+μ^2t). Since
sup {-t^-1sin t: t<0}<1/4, we deduce that there exists μ <1/2 
such that the curve γ is self-expanded. □

Notation. Throughout this subsection, γ will refer to the curve given in
Lemma <ref> and μ<1/2 will be the constant fixed there.

The following lemma says that the curve γ constructed in the previous lemma  
satisfies that for each τ, the associated cone
C(t,α), α=arccos(1/√(5)) does not meet the z-axis, that is,
the axis of evolution of the spiral curve.



Let γ:ℝ→ℝ^3 be a spiral of the
form (<ref>). If λ=1/√(5) and α=arccos(λ), 
then the cone C(t,α) does not intersect the line
parametrized by ℓ(z)=(0,0,z).


Proof. Under the notation of the previous lemma, it is
enough to verify that for all z∈ℝ

    ⟨γ'(0),ℓ(z)-γ(0)⟩≤1/√(5) ‖γ̇(0)‖ ‖ℓ(z)-γ(0)‖ .

The above condition reads

    μ rz≤1/√(5) √(r^2(1+μ^2)) √(r^2+z^2),    for
    all z∈ℝ,

or equivalently,

    (z/r)/√(1+(z/r)^2)≤ √((1+μ^2))/μ√(5),    for all z∈ℝ.

Since t↦ t^-1√(1+t^2) is decreasing for t>0 and μ<1/2,
we have μ^-1√(1+μ^2)>√(5), therefore (<ref>) is
satisfied.□



We shall now enhance in the above construction to deduce that
the cone C(τ,α) avoids a thin (infinite)
cylinder

    Cyl (r_0)={(x,y,z)∈ℝ^3; x^2+y^2=r_0
    ^2, z∈ℝ}

containing the z-axis. Indeed, taking r_0<<r the above cylinder
is very close to the z-axis, therefore we obtain (almost) the same result as
before. This is formulated in the next lemma.


 There exists an integer N≥2 such that  
whenever r=Nr_0 and α=arccos1/√(5), we
have:

    C(α,τ)∩Cyl (r_0)=∅,   for all τ≥0.




Proof. We consider again the curve γ given by
(<ref>). Thanks to the symmetry, it is enough to check the assertion for
τ=0. Therefore, for σ(θ,z)=(  r_0cosθ,r_0sinθ,z), it is enough to verify

    ⟨γ'(0),σ(θ,z)-γ(0)⟩≤cosα ‖γ'(0)‖ ‖σ
    (θ,z)-γ(0)‖    ∀θ∈0,2π], ∀
    z≥0,

where γ(0)=(  r,0,0)and γ'(0)=(  0,r,μ
r)  .The above condition reads

    rr_0sinθ+μ rz≤cosα √(r^2(1+μ^2)) √((
    r_0cosθ-r)  ^2+r_0^2sin^2θ+z^2)  ∀θ∈0,2π], ∀ z≥0.

Dividing by rr_0, setting w=z/r_0, and since cosα=1/√(5), we
deduce

    sinθ+μ w≤1/√(5) √(1+μ^2) √((  r/r_0-1)  ^2+2r/r_0(1-cosθ)+w^2).

Setting r=Nr_0 we obtain the condition

    1/√(5)≥1/√(1+μ^2) sup_θ∈0,2π
    ],w∈ℝ{sinθ+μ|w|/√((N-1)^2
    +2N(1-cosθ)+w^2)}  .

But for any θ∈0,2π], u=|w|≥0

    sinθ+μ u/√((N-1)^2+2N(1-cosθ)+u^2)≤1+μ u/√((N-1)^2+u^2)

and

    sup_u≥0{1+μ u/√((N-1)^2+u^2)}
    =1/N-1 √(1+μ^2(N-1)^2)⟶μ    as
       N→+∞.

Since  μ(  1+μ^2)  ^-1/2<(  √(5))  ^-1,
we can choose N large enough such that

    √(1+μ^2(N-1)^2)/(N-1)√(1+μ^2)<1/√(5).

Therefore, for this choice of N, we get
C(0,α)∩Cyl (r_0)=∅. □





Let γ be given by (<ref>). We shall now include a further
restriction. We shall show that the cone C(τ,α) associated to γ also
avoids radial segments S of the form:

    S={(x,0,0); 0≤ x≤ r}.

This is the aim of the following lemma.


 If λ=1/√(5) and α=arccos(λ),
then C(τ,α)∩ S=∅ for all τ≥0.


Proof. It is enough to verify

    ⟨γ'(τ),(x,0,0)-γ(τ)⟩≤cosα ‖γ̇(τ)‖ ‖ (x,0,0)-γ
    (τ)‖ ,    for all  0≤ x≤ r,

where

    γ(τ)=(  r cosτ, r sinτ, μ rτ)
        γ̇(τ)=(  -rsinτ, rcosτ, μ
    r)  .

Setting λ=cosα and simplifying by r, we obtain for all 0≤
x≤ r

    -xsinτ-μ^2rτ ≤ λ √(1+μ^2) √((x-rcosτ)^2+r^2sin^2τ+μ^2r^2τ^2),



Notice that μ satisfies sinτ+μ^2τ>0 for every τ≥0. Therefore,

    -xsinτ-μ^2rτ≤-x(sinτ+μ^2τ)≤0,

so (<ref>) is clearly satisfied.□





 §.§ Arbitrary long eels inside a bounded cylinder


We are now ready to construct arbitrarily long λ-eels lying inside
the following bounded cylinder:

    Cyl (r,[a,a+2πμ r]):={(x,y,z)∈ℝ^3; x^2
    +y^2=r^2, a≤ z≤ a+2πμ r}.

Indeed we have the following result.


 Let λ≥1/√(5) and
let Cyl (r,[a,a+2πμ r]) be the bounded cylinder defined in
(<ref>). Then there exists a λ-eel

    γ:I⟼Cyl (r,[a,a+2πμ r])

whose length is greater than 1. Moreover, the initial point of γlies in the upper part of the cylinder (z=a+2πμ r) while the last
point lies at the bottom (z=a).


Proof. Without loss of generality, we assume a=0.
Below, N is a fixed integer given by Lemma <ref>. Let us
fix an odd integer n such that 2πμ rn>1. Then for 1≤ k≤ n, we
define internal cylinders

    C_k:=Cyl (r/N^n-k,[0,2πμ r])={(x,y,z)∈ℝ^3; x^2+y^2=(  r/N^n-k)  ^2, 0≤
    z≤2πμ r}.

For k=2ℓ+1≤ n (odd) we define a downward spiral curve γ
_k^↓ as follows:

    γ_k^↓(t)=r/N^n-k(
    cos(t), sin(t), μ(2π N^n-k-t)),     for 0≤ t≤2π
    N^n-k.

while for k=2ℓ≤ n (even) we define an upward spiral curve γ
_k^↑ as follows:

    γ_k^↑(t)=r/N^n-k(
    cos(t), sin(t), μ t),    for 0≤ t≤2π N^n-k.

Notice that if k odd,

    γ_k^↓(0)=(r/N^n-k,0,2πμ r)  and  γ_k^↓(2π N^n-k)=(r/N^n-k,0,0),

while for k even

    γ_k^↑(0)=(r/N^n-k,0,0)  and  γ
    _k^↑(2π N^n-k)=(r/N^n-k,0,2πμ r).

Each spiral γ_k lies on the surface of the cylinder C_k and makes
N^n-k loops to reach the upper part of the cylinder starting from the
bottom and going upwards if k is even (respectively, to reach the bottom,
starting from the upper part and going downward, if k is odd). We finally
define parametrized segments e_k^+ joining the end point of γ_k^↓ 
to the initial point of γ_k^↑ (for k=2ℓ+1), and respectively e_k^- 
joining the end point of γ_k^↑ to the initial point of γ_k+1^↓ (for
k=2ℓ), that is:

    e_k^+(t)=(r/N^n-k(1+t(N-1)),0,2πμ r)
      and  e_k^-(t)=(r/N^n-k(1+t(N-1)),0,0)  ,   t∈0,1].





The curve γ will now be defined concatenating the above curves: we
start with k=1 and the downward spiral γ_1^↓ and we
concatenate with the segment e_1^-. We continue with the upward spiral
γ_1^↑ and the segment e_1^+ and concatenate with
γ_2^↓ (k=2), then the segment e_2^- and so on, up
to the final downward spiral γ_n^↓. The resulting curve is
clearly continuous. Applying Lemma <ref> and Lemma <ref>
we deduce that γ is a λ-eel. The length of γ is clearly greater than 1
since we cross n times the cylinder of length 2πμ r (and we have taken n≥2 such that 2πμ rn>1). □




 Proposition <ref> ensures, by rescaling, that we can construct
arbitrarily long λ-eels inside arbitrarily small cylinders. The
λ-eel γ inside the cylinder
Cyl (r,[a,b]), where b=a+2πμ r, 
is obtained by concatenating pieces of
three different types:

- Type 1: a spiral going downward: γ_i^↓(t)=(ρcos(t),ρsin(t),b-ρμ t),

- Type 2: a spiral going upward: γ_i^↑(t)=(ρcos(t),ρsin(t),a+ρμ t),

- Type 3: a segment parametrized by e_i^-(t)=(t,0,a) or
by e_i^+(t)=(t,0,b).



It is possible to modify slightly the above construction to get a λ-ell (with λ=1/√(5))
γ:(-∞,0]→Cyl (r,[a,a+2πμ r]), with infinite length,
and such that lim_t→ -∞γ(t) does not exist.
Since the curve constructed γ above depends on the parameter n,
let us denote it γ_n. We can assume, without loss of generality, by choosing a suitable parametrization, 
that γ is defined on [-n,0] and that for each n,
the restriction of γ_n+1 to  [-n,0] coincides with γ_n.
Now we define γ on (-∞,0], satisfying, for each n, γ_|[-n,0]=γ_n.
Since each γ_n is a λ-ell, it is clear that γ is  a λ-ell.
Morover, the z-coordinate of γ(t) oscillates infinitely many times between a and a+2πμ r.
This shows both that γ has  infinite length
and that lim_t→ -∞γ(t) does not exist.





 §.§ Constructing bounded eels of infinite length in 3D


To construct a bounded λ-eel with infinite length, we need to glue
together curves of length greater than 1 (constructed in the previous
subsection) that lie each time in prescribed disjoint bounded cylinders, all
taken along the z-axis, of the form C_n:=Cyl (r_n
,[a_n,b_n]) with a_n>b_n+1 and r_n↘0^+. To construct
efficiently such a curve, and to establish that it is a λ-eel, we
shall need the following result, asserting that a λ-eel lying in a
small cylinder does not see a bigger remote cylinder of the same axis.


 Let λ=1/√(5), α=arccos(λ),
and let us set

    Cyl (R,[a,b]):={(x,y,z)∈ℝ^3; x^2+y^2≤
    R, a≤ z≤ b}.

Then there exists M>1 such that, for every r∈(0,R/2) and a^',b^'∈ℝ such that

    a^'<b^'<a<b   and   b^'-a^'≥ MR

the curve γ:[a^',b^']→ℝ^3
with equation γ(t)=(  r cos t, r sin t, μ rt),
satisfies

    C(τ,α)∩Cyl (R,[a,b])=∅.




Proof. Without loss of generality, we can assume b^'=0. The equation of the spiral γ:[0,∞)→ℝ
^3 is of the form

    γ(t)=(  r cos t, r sin t, μ rt)  ,     a^'≤ t≤0,

where μ<1/2 is given by Lemma <ref>. 
Set

    σ(θ,z,u)=(  ucosθ,usinθ,z)  ,  γ(0)=(  r,0,0)      γ'(0)=(
    0,r,μ r)  .

It is enough to check that

    ⟨γ'(0),σ(θ,z,u)-γ(0)⟩≤cosα ‖γ'(0)‖ ‖σ
    (θ,z,u)-γ(0)‖    ∀θ∈0,2π
    ], ∀ z∈ a,b], ∀ u∈0,R].

The above condition reads, for all θ∈0,2π], for all z∈ a,b] and for all u∈0,R],

    rusinθ+μ rz≤1/√(5) √(r^2(1+μ^2)) √((
    ucosθ-r)  ^2+u^2sin^2θ+z^2).

So it is enough to check that for all z∈
a,b] and u∈0,R] it holds

    u+μ z≤1/√(5) z√(1+μ^2).

In order to do that, let us fix the value of M. 
For μ<1/2, we have √(1+μ^2)>√(5)μ.
Therefore, we can choose M>0 such that √(1+μ^2)>(  μ+1/M)  √(5).
Now for all u≤ R and z≥ a we
have

    u+μ z/z≤R/a+μ≤1/M+μ
    <√(1+μ^2)/√(5),

This completes the proof of the lemma. □



We are now ready to prove Theorem <ref>, that is, given λ
=1/√(5), we construct a continuous curve γ
:[0,+∞]→ℝ^3 of infinite length, lying in the unit
ball, with nonzero right derivative at each point and satisfying the λ-cone property (λ-eel).




Proof of Theorem <ref>.
We claim that we can construct a sequence of disjoint bounded cylinders

    C_n=Cyl (r_n,[a_n,a_n+2πμ r_n]),   n≥ 1
 
along the z-axis, such
that a_n∈0,1), r_n+1≤ r_n/2,
ℓ_n:=a_n-(a_n+1+2πμ r_n+1)>0 and ℓ_n/r_n is sufficiently
big to ensure that the cylinder C_n is not seen by any λ-eel lying
in a (smaller) cylinder C_m for m>n (c.f. Lemma
<ref>). More precisely, we define a_0=0, and, for n≥1,

    a_n=2^-n   and  r_n=1/2^n+1(πμ+M),

where M>0 is given by Lemma <ref>. Let us check that the
conditions of Lemma <ref> are fulfilled for the cylinders C_n
(big remote cylinder) and C_n+1 (small cylinder):

    ℓ_n=a_n-(a_n+1+2πμ r_n+1)=1/2^n+1-2πμ/2^n+2(πμ+M)=M/2^n+1(πμ+M)≥ Mr_n.

Now the construction is as follows. For each n, let γ_n be the
λ-eel given by Proposition <ref>, of length greater than 1 lying
inside the cylinder C_n, entering this cylinder from the upper part
(z=a_n+2πμ r_n) and having its endpoint at the bottom (z=a_n).
Let ẽ_n be the oriented segment going from the endpoint of the
curve γ_n (bottom of the cylinder C_n) to the starting point of
γ_n+1 (upper part of the cylinder C_n+1). We now define
γ:[0,+∞)→ℝ^3 by concatenation of the
following curves : γ_1, ẽ_1, γ_2, ẽ
_2, and so on. It is clear that γ is continuous and has right
derivative at each point. Morever, γ is contained in the unit ball of
ℝ^3 and its length ℓ(γ) is greater than ℓ
(γ_1)+ℓ(γ_2)+⋯+ℓ(γ_n)≥ n for every n,
therefore it is infinite. Observe that γ(t) has limit 0 as
t→+∞. It remains to prove that γ is a λ-eel,
that is, it satisfies the λ-cone condition. Notice that each curve
γ_n, ẽ_n is individually a λ-eel (that is, it satisfies the λ-cone
property with respect to itself). Provided M is sufficiently big, the segment ẽ_n 
is almost parallel to the z-axis and it is oriented to the opposite direction of the previous curves γ_1,
ẽ_1,⋯, γ_n. Therefore, if the λ-cone C(t,α), given in (<ref>), has its origin onto a segment ẽ_n, then it does not meet the union of the ranges of γ_1, ẽ_1,⋯, γ_n. It remains to treat the case where C(t,α) has its origin to a curve of the form γ_n. These curves are constructed (for each n) by concatenating pieces of the form γ_i^↓ (of type 1), γ
_i^↑ (of type 2) and e_i^+ or e_i^- (of type 3) (c.f. Remark <ref>).
If the λ-cone lies on a piece of type 1 or of type 3 of γ_n, then it is oriented to the opposite directions of all of the previous pieces γ_1, ẽ_1,⋯, γ_n-1, ẽ_n-1 of γ, therefore it does not meet the union of their ranges. If now the cone C(t,α) has its origin on an upward piece γ_i^↑ (type 2) of the curve γ_n, then the result follows
from Lemma <ref>. The proof is complete. □





§ CURVES WITH THE Λ-CONE PROPERTY IN 2 DIMENSIONS


It is remarkable that there is no analogue of the construction in Theorem <ref> in dimension 2. Indeed, we shall show that for any value of the parameter λ∈ -1,1), any bounded planar λ-eel
(that is, continuous curve with right derivative at each point that satisfies the λ-cone property) is rectifiable and has finite
length. We shall need the following lemmas. (Recall α =arccos(λ ).)


Let γ :I→ℝ^2 be a planar 
λ-eel and t_1<t_2<t_3 in I. Then 

    γ (t_3)∉γ (t_1),γ (t_2)].



Proof. Set A=γ (t_1), B=γ (t_2), C=γ (t_3) and assume towards a contradiction that C∈ A,B]. Choosing adequate coordinates in ℝ^2 we may assume that 
A=(0,0), B=(1,0) and C=(c,0) with c∈ (0,1). In the sequel, we shall
write γ =(γ _1,γ _2) in these coordinates.

Before we proceed, notice that we may assume

    γ (t)∉ (A,C)   for all t∈ (t_1,t_2].

Indeed, set N_1={t∈ t_1,t_2): γ (t)∈
A,C]}={t∈ t_1,t_2): γ _1(t)∈ 0,c], γ
_2(t)=0} and α _1:=sup{γ _1(t):t∈ N_1}. Then 
α _1<c (since γ is continuous and injective) and
consequently, there exists t_1≤t̃_1<t_2 with γ (t̃_1)=(α _1,0)=Ã. In this case we can replace A
by Ã and t_1 by t̃_1 and get (<ref>).

We set t̃_2=inf{t∈ t_1,t_2]: γ _1(t)≥
c, γ _2(t)=0}. There is no loss of generality to assume t_2=
t̃_2, since we can always replace B by B̃=(γ _1(t̃_2),0) (notice that γ _1(t̃_2)>c by
injectivity).

Therefore for all t∈ (t_1,t_2) we have γ (t)∉ (A,B).
Setting Γ _AB={γ (t):t∈ t_1,t_2]} we deduce
that Γ _AB∪ (B,A] is a Jordan curve which separates ℝ
^2 in two regions, exactly one of them being bounded. Call ℛ
this bounded region, set H^+={x=(x_1,x_2):x_2>0}, H^-={x=(x_1,x_2):x_2<0} and let ε >0 be such that B(C,ε )∩Γ _AB=∅ . Then at least one of the
sets B(C,ε )∩ H^+ and B(C,ε )∩ H^- has
nonempty intersection with ℛ. Assume, with no loss of
generality, that

    B(C,ε )∩ H^-∩ℛ≠∅ .

Then for every x∈ H^-∩int ℛ and every
direction d=(d_1,d_2)∈𝕊^1 (the unit sphere of ℝ
^2) with d_2≤ 0 it holds ℓ _x,d∩Γ _AB≠∅ , where ℓ _x,d:={x+μ d:μ≥ 0} is the half-line
emanating from x with direction d. In particular, shrinking ε >0
if necessary, and recalling notation (<ref>) we deduce that 

    C_x(d,α )∩Γ _AB≠∅ ,  for all x∈
    B(C,ε )∩ H^-∩ℛ and all 
    d=(d_1,d_2) with d_2≤ 0.

Let τ _3∈ (t_2,t_3) be such that for all t∈ (τ _3,t_3] we have γ (t)∈ B(C,ε ) (such τ _3 exists by
continuity). Then it follows by (<ref>) and the λ-eel property
that γ'_2(t)>0, and consequently, γ _2(t)<0 (since γ _2(t_3)=0). Let further τ∈ t_2,t_3] be such
that

    γ _2(τ )=min_t∈ t_2,t_3]γ _2(t) ( <0 ).

Then since γ _2(t_2)=0, there exists t̃∈
t_2,τ ] with (γ (t)∈ℛ and) γ'_2(t)<0
which together with (<ref>) contradicts the λ-eel
property.□



For the next statement, recall notation (<ref>) and (<ref>).


Under the assumptions of the previous lemma we have: 

    C(γ'(t),α)∩ K(t)={0},  for all t∈ I.



Proof. Fix t∈ I and assume with no loss of generality
(by translation) that γ (t)=0. Then K(t)=cone (Γ (t)-γ (t))=cone Γ
(t), where Γ (t)={γ (τ ):t∈ 0,t]}. Let assume
that there exists x∈ C(γ'(t),α)∩ K(t), x≠ 0. Then by
Caratheodory theorem, there exist x_i=γ (τ _i), i∈{1,2,3} with τ _1≤τ _2≤τ _3<t and x∈conv{x_1,x_2,x_3} (convex envelope). Set ℓ _1:={x_1+μ
(x-x_1): μ≥ 0}  and ℓ _2={x_2+μ (x-x_2): μ≥
0}. If ℓ _1∩Γ (t)=ℓ _2∩Γ (t)=∅ ,
then for μ _1, μ _2 sufficiently big, the point x_3 should
belong to the triangle defined by the points ℓ _1(μ _1):=x_1+μ
_1(x-x_1),  x and ℓ _2(μ _2)=x_2+μ _2(x-x_2). Then
by connectedness of γ ([τ _1,τ _3]), we deduce that for
some s<τ _3<t it holds γ (s)∈ℓ _1∪ℓ _2. We
deduce that x is a convex combination of two points of Γ (t), that
is, x∈γ (s_1),γ (s_2)] for some s_1<s_2<t.
Set Γ _12:={γ (τ}:τ∈ s_1,s_2]}. Since 
γ is a λ-eel, we have C(t,α)∩Γ _12=∅ .
Then [γ (s_2),γ (s_1)]∪Γ _12 is a Jordan curve
and γ (t)=0∈ℛ where ℛ is the bounded region
delimited by the Jordan curve. This yields that for some t_1<t_2<t, γ (t)=0 is a convex combination of γ (t_1) and γ(t_2), which contradicts Lemma <ref>.□



In view of Lemma <ref> and Remark <ref> we obtain
our main result.


Let γ :I→ℝ^2 be a bounded 
λ-eel. Then γ is rectifiable and has finite length.□




Acknowledgment. The authors wish to thank Ludovic Rifford for several useful
discussions.
99                                                                                               


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Aris Daniilidis



DIM–CMM, UMI CNRS 2807Beauchef 581, Torre Norte, piso 5,
Universidad de ChileSantiago CP8370456, Chile 

E-mail:  



Research supported by the grants: BASAL PFB-03, FONDECYT
1171854, ECOS/CONICYT C14E06, REDES/CONICYT 15040 (Chile) and
MTM2014-59179-C2-1-P (MINECO of Spain and ERDF of EU).




Robert Deville



Laboratoire Bordelais d'Analyse et GeométrieInstitut de
Mathématiques de Bordeaux, Université de Bordeaux 1351 cours
de la Libération, Talence Cedex 33405, France 

E-mail: 



Research supported by the grants: ECOS/CONICYT C14E06
(France) and REDES/CONICYT-15040 (Chile).




Estibalitz Durand-Cartagena



Departamento de Matemática AplicadaETSI Industriales,
UNEDJuan del Rosal 12, Ciudad Universitaria, E-28040 Madrid, Spain


E-mail:  



Research supported by the grant MTM2015-65825-P (MINECO of Spain) and 2018-MAT14 (ETSI Industriales, UNED).


