SELF-CONTRACTED CURVES IN SPACES WITH WEAK
LOWER CURVATURE BOUND
NINA LEBEDEVA, SHIN-ICHI OHTA, AND VLADIMIR ZOLOTOV
Abstract. We show that bounded self-contracted curves are rectifiable
in metric spaces with weak lower curvature bound in a sense we introduce
in this article. This class of spaces is wide and includes, for example,
finite-dimensional Alexandrov spaces of curvature bounded below and
Berwald spaces of nonnegative flag curvature. (To be more precise, ourcondition is regarded as a strengthened doubling condition and holds also
for a certain class of metric spaces with upper curvature bound.) We also
provide the non-embeddability of large snowflakes into (balls in) metricspaces in the same class. We follow the strategy of the last author’s
previous paper based on the small rough angle condition, where spaces
with upper curvature bound are considered. The results in this article
show that such a strategy applies to spaces with lower curvature boundas well.

1. Introduction
1.1. Self-contracted curves. Let (X, d) be a metric space and I ⊂ R aninterval. A curve γ : I −→ X is said to be self-contracted if()()
d γ(t2 ), γ(t3 ) ≤ d γ(t1 ), γ(t3 )holds for every t1 ≤ t2 ≤ t3 (we remark that γ is not necessarily continuous).
This notion was introduced in [10] and turned out quite useful for studying
the rectifiability (finiteness of the length) of γ. Self-contracted curves arise as
gradient curves of quasi-convex functions (see Definition 29) in “Riemannianlike” metric spaces (see Section 6), and the rectifiability of gradient curves
is an intriguing problem in optimization theory and dynamical systems. We
refer to [8,10] for more about the background information on self-contracted
curves, and [10, Section 6] for further examples of self-contracted curves.
Recall that a curve γ : I −→ X is said to be rectifiable if
}
{∑
k−1
(
)
sup
d γ(ti ), γ(ti+1 ) t1 , . . . , tk ∈ I, t1 < · · · < tk < ∞.
i=1

2010 Mathematics Subject Classification. 51F99.
Key words and phrases. isometric embedding, self-contracted curve, snowflake.
1

It was shown that bounded self-contracted curves are rectifiable, on the one
hand, in the following spaces:• Euclidean spaces [8, 21],
• Riemannian manifolds [9],• finite-dimensional normed spaces [20, 35],
• CAT(0)-spaces with some additional properties [30].
These results are more or less based on the fundamental construction in the
Euclidean case. Precisely, such a construction was generalized to normed
spaces and CAT(0)-spaces, and the case of Riemannian manifolds was shownby approximations and a compactness argument.
On the other hand, an observation in [40] revealed an interesting connection between self-contracted curves and an embedding problem of finite
metric spaces, and the following spaces were added to the above list:• complete Finsler manifolds,• complete, locally compact CAT(k)-spaces (k ∈ R) with locally extendable geodesics,• complete, locally compact Busemann NPC spaces with locally extendable geodesics,• complete, locally compact geodesic spaces with locally extendable
geodesics satisfying the local rough Busemann condition(see [40] for the precise meaning of those conditions on geodesics and curvature bounds). Let us stress that all the above results are on spaces withupper curvature bounds.
The aim of the present paper is to show that the strategy in [40] also
applies to spaces with lower curvature bounds. As a result we further extend
the above list with:• finite-dimensional Alexandrov spaces of curvature ≥ k with k ∈ R(Proposition 14),• complete Berwald spaces of nonnegative flag curvature (Proposition 17).
These are consequences of our main result below (Theorem 2). Our approachalso provides an alternative proof for normed spaces. We remark that thefiniteness of the dimension is essential since one can easily construct counterexamples in Hilbert spaces (see [9, Example 2.2], [30, Example 2.6]).
1.2. Small rough angle condition. As in [40], the following condition
inspired by [18] will play an essential role.
Definition 1 (Small rough angle condition; SRA(α)). Let (X, d) be a metricspace and 0 < α < 1. We say that X satisfies the SRA(α)-condition if, forevery x, y, z ∈ X, we haved(x, y) ≤ max{d(x, z) + αd(z, y), αd(x, z) + d(z, y)}.
2

(1.1)

Note that (1.1) implies, provided x, y ∈ X \ {z},e∠xzy
≤ arccos(−α) = π − arccos(α),

(1.2)

e
where ∠xzy
∈ [0, π] denotes the Euclidean comparison angle defined as theangle ∠x̃z̃ ỹ for the comparison triangle △x̃ỹz̃ ⊂ R2 having the same side
lengths as △xyz ⊂ X. By the Euclidean cosine formula we haved(x, z)2 + d(y, z)2 − d(x, y)2ecos ∠xzy=.
(1.3)
2d(x, z)d(y, z)eIndeed, if the opposite inequality ∠xzy
> π − arccos(α) to (1.2) holds, thenwe haveee
α < cos(π − ∠xzy)= − cos ∠xzyand (1.3) impliesd(x, y)2 > d(x, z)2 + d(y, z)2 + 2αd(x, z)d(y, z)> (max{d(x, z) + αd(z, y), αd(x, z) + d(z, y)})2 .
In [18] the condition SRA(α) was used to study the non-embedding ofsnowflakes (see the next subsection). A connection with self-contracted
curves was established in [40, Theorem 1] as follows.
Theorem 1 (SRA-sets in self-contracted curves, [40]). Assume that a metricspace (X, d) admits a bounded, self-contracted, and unrectifiable curve γ :I −→ X. Then, for any 1/2 < α < 1 and k ∈ N, there exists a k-point
subset Y ⊂ γ(I) satisfying the SRA(α)-condition.
We remark that, if γ(s) = γ(t) for s < t, then γ is constant on [s, t] by the
self-contractedness (see [30, Lemma 2.1]). Thus the unrectifiability implies
that the image of γ is indeed an infinite set. Theorem 1 asserts that, inother words, if a metric space does not admit any large subset satisfying
SRA(α), then every bounded self-contracted curve in it is rectifiable. Alongthis strategy of [40], we introduce metric spaces with weak lower curvaturebound and show that they do not contain large SRA(α)-subsets.
1.3. Angular total boundedness and main theorems. We denote by
BR (x) the open ball of center x and radius R.
Definition 2 (Angular total boundedness; ATB(ϵ)). Let (X, d) be a metric space and 0 < ϵ < π/2. We say that a point p ∈ X satisfies the
ATB(ϵ)-condition if there exist some L ∈ N and R > 0 such that, for every
y1 , . . . , yL ∈ BR (p) \ {p}, we can find i ̸= j satisfying
e i pyj < ϵ.
∠yWe say that X is a globally ATB(ϵ)-space if there exists L ∈ N such thatevery p ∈ X satisfies the ATB(ϵ)-condition with constants L and R = ∞.
3

We say that X is a semi-globally ATB(ϵ)-space if, for any x ∈ X and R > 0,
there exists L ∈ N such that every p ∈ BR (x) satisfies the ATB(ϵ)-condition
with constants L and R. Finally we say that X is a locally ATB(ϵ)-space
if, for any x ∈ X, there exist L ∈ N and R > 0 such that every p ∈ BR (x)satisfies the ATB(ϵ)-condition with constants L and R.
Roughly speaking, the ATB(ϵ)-condition asserts the total boundedness
with respect to the comparison angle. Examples of locally ATB(ϵ)-spaces
include:• Finsler manifolds (Proposition 16),• complete, locally compact CAT(k)-spaces (k ∈ R) with locally extendable geodesics (Proposition 12(1)).
Next we have semi-globally ATB(ϵ)-spaces:• finite-dimensional Alexandrov spaces of curvature ≥ k with k ∈ R(Proposition 14),• complete, locally compact Busemann NPC spaces (e.g., CAT(0)spaces) with locally extendable geodesics (Proposition 12(2)).
Finally, globally ATB(ϵ)-spaces include:• finite-dimensional Alexandrov spaces of nonnegative curvature (Proposition 14),• finite-dimensional normed spaces (Proposition 15),• complete Berward spaces of nonnegative flag curvature (Proposition 17),
• Cayley graphs of virtually abelian groups (Proposition 18).
Notice that the class of ATB(ϵ)-spaces is so large that it admits both lower
and upper curvature bounds, as well as both Riemannian and Finsler spaces.
Our main theorem is the following.
Theorem 2 (ATB prevents large SRA-sets). Let (X, d) be a metric space,0 < α < 1 and ϵ := arccos(α)/2 > 0. Given x ∈ X and R > 0, supposethat there is L ∈ N such that every p ∈ BR (x) satisfies the ATB(ϵ)-condition
with constants L and R > 0. Then there exists N (L) ∈ N such that there is
no N -point SRA(α)-subset in BR (x).
We give the proof of Theorem 2 in Section 2. Combining Theorem 2 with
Theorem 1, we obtain the following. A metric space is said to be proper ifevery bounded closed set is compact.
Theorem 3 (Rectifiability of self-contracted curves). Let 0 < ϵ < π/6. If(X, d) is a semi-globally ATB(ϵ)-space or a proper locally ATB(ϵ)-space, thenevery bounded self-contracted curve in X is rectifiable.
The following aforementioned results are now given as a corollary.
4

Corollary 3. Bounded self-contracted curves in finite-dimensional Alexandrov spaces of curvature ≥ k or in complete Berwald spaces of nonnegative
flag curvature are rectifiable.
1.4. Non-embeddability of snowflakes. The SRA(α)-condition finds applications in the embedding problem via the following observation foundin [18, (3.3)]. Given a metric space (Y, d) and 0 < α < 1, we denote by(Y, dα ) its α-snowflake defined by dα (x, y) := d(x, y)α .
Proposition 4 (Snowflakes satisfy SRA, [18]). For any metric space (Y, d)and 0 < α < 1, the α-snowflake (Y, dα ) satisfies the SRA(α)-condition.
Combining this with Theorem 2 leads the following.
Theorem 4 (ATB prevents large snowflakes). Let (X, d) be a metric space,0 < α < 1 and ϵ := arccos(α)/2. Given x ∈ X and R > 0, suppose that
there is L ∈ N such that every p ∈ BR (x) satisfies the ATB(ϵ)-condition withconstants L and R > 0. Then there exists N = N (L) ∈ N such that, if ametric space (Y, dY ) has cardinality at least N , then its α-snowflake (Y, dαY )does not admit an isometric embedding into BR (x).
As a corollary, we obtain restrictions on the isometric embeddability oflarge snowflakes into Alexandrov spaces or Berwald spaces of nonnegative
flag curvature. Here we state the former for completeness.
Corollary 5. Let (X, d) be an n-dimensional Alexandrov space of curvature≥ k with k ∈ R. Then, for any 0 < α < 1 and R > 0, there exists
N = N (n, k, α, R) ∈ N such that, if a metric space (Y, dY ) has cardinality atleast N , then its α-snowflake (Y, dαY ) does not admit an isometric embedding
into any R-ball in X. Moreover, if k = 0, then we can take R = ∞ and
N = N (n, α).
This article is organized as follows. In Section 2, we prove Theorems 2 and3. In Section 3, we introduce a technical version of the ATB(ϵ)-condition
named the ATB∗ (ϵ)-condition, which is useful for proving ATB(ϵ). With
the help of the tools introduced in Section 4, we give examples of locally,semi-globally, and globally ATB(ϵ)-spaces in Section 5. Section 6 is devoted
to the self-contractedness of gradient curves of quasi-convex functions. In
Section 7, we show that the absence of large SRA(α)-subsets implies the
doubling condition, as a weak converse to Theorem 2. Finally, in Sections 8
and 9, we discuss several intriguing (counter-)examples and open questions,
respectively.
2. Proofs of Theorems 2 and 3
The proof of Theorem 2 makes use of the Ramsey theory. For a, b ∈ N, wedenote by R(a, b) the corresponding Ramsey number. That is to say, when5

we color the edges of the complete graph of R(a, b)-vertices in two colors,then there exists either an a-point subset of vertices such that all the edges
between these vertices are in the first color, or a b-point subset of vertices
such that all the edges between these vertices are in the second color.
Proof of Theorem 2. The hypotheses of the theorem means that, for everyp ∈ BR (x) and y1 , . . . , yL ∈ BR (x) \ {p}, there exists 1 ≤ i < j ≤ L suchthat
e i pyj < ϵ.
∠y(2.1)
Let us define
HL := R(2, L),

HL−1 := R(HL + 1, L),...,

H2 := R(H3 + 1, L).

We shall show that we can take N = H2 + 1. By contradiction let Z be an
N -point SRA(α)-subset in BR (x). Take a point z1 from Z and consider thecomplete graph of the remaining H2 points. We color edges of the graph intwo colors. We color an edge {v, w} in the first color if
e 1 w < ϵ,∠vz
and in the second otherwise. By (2.1) there is no L-point subset such that
all edges in the corresponding subgraph are of the second color. Thus, by
the choice of H2 = R(H3 + 1, L), there exists an (H3 + 1)-point subset
such that all edges in the corresponding subgraph are of the first color. We
denote this set of (H3 + 1) points by Ze2 , and take some z2 ∈ Ze2 . Next we
e 2 w < ϵ or not, findset Z2 := Ze2 \ {z2 } and consider the coloring by ∠vzan (H4 + 1)-point set Ze3 ⊂ Z2 and take z3 ∈ Ze3 in the same way. Put
Z3 := Ze3 \ {z3 } and iterate this procedure. In the last step we find a 2-point
set ZeL and let us denote it by ZeL = {zL , zL+1 }.
e j zi zk < ϵOn the one hand, by the definition of the first color we have ∠z
for any 1 ≤ i < j < k ≤ L + 1. This implies thate i zk zj + ∠z
e i zj zk = π − ∠z
e j zi zk > π − ϵ.
∠z
On the other hand, we deduce from the SRA(α)-condition (1.2) thate i zj zk ≤ π − arccos(α) = π − 2ϵ.
∠zCombining the last two inequalities provides
e i zk zj > ϵ∠zfor every 1 ≤ i < j < k ≤ L + 1. Therefore, if we take p = zL+1 and yi = zi
for i = 1, . . . , L, then we have a contradiction with (2.1). This completes theproof.
□
Theorem 3 is almost straightforward, here we give a proof for completeness.
6

Proof of Theorem 3. The semi-global case is immediate, given a bounded,self-contracted and unrectifiable curve γ, one can find an arbitrary large
SRA(α)-set in the image of γ with α = cos(2ϵ) ∈ (1/2, 1) by Theorem 1.
This contradicts Theorem 2.
In the local case, let us consider the covering {BRx (x)}x∈X of X, where
every p ∈ BRx (x) satisfies the ATB(ϵ)-condition for some Lx ∈ N and Rx > 0.
Since X is proper and γ is bounded, the image of γ is covered by finite
elements of {BRx (x)}x∈X . Then, if γ is not rectifiable, the restriction of γ toone of those balls is unrectifiable. This implies a contradiction similarly tothe previous argument.
□
3. ATB∗ (ϵ)-condition
The aim of this section is to introduce a technical version of the ATB(ϵ)condition, named the ATB∗ (ϵ)-condition. The following lemma connects
these conditions. For 0 < ϵ < π/2, we defineβ(ϵ) :=(1 − cos ϵ) sin ϵ.
2(1 + sin ϵ)(3.1)

Notice that β(ϵ) < 1/4 and β(ϵ) = O(ϵ3 ) as ϵ → 0.
Lemma 6. Let (X, d) be a metric space, and take p ∈ X and x, y ∈ X \ {p}.
If there is x′ ∈ X such that d(p, y) = d(p, x′ ) + d(x′ , y) and d(x, x′ ) ≤eβ(ϵ)d(p, x), then we have ∠xpy
< ϵ.
Proof. First of all, if d(x, y) < d(p, x) sin ϵ, then we find (recalling (1.3))ecos ∠xpy=d(p, x)2 + d(p, y)2 − d(x, y)2
cos2 ϵ · d(p, x)2 + d(p, y)2>.
2d(p, x)d(p, y)2d(p, x)d(p, y)eeThus Young’s inequality shows cos ∠xpy
> cos ϵ and hence ∠xpy
< ϵ (with′out the hypothesis on the existence of x ).
Assume d(x, y) ≥ d(p, x) sin ϵ (and hence x ̸= y). We deduce from thetriangle inequality and the hypotheses thatd(p, y) = d(p, x′ ) + d(x′ , y) ≥ d(p, x) + d(x, y) − 2d(x, x′ )≥ d(x, y) + (1 − 2β)d(p, x)
with β = β(ϵ) for simplicity. Substitutinged(p, y)2 = d(p, x)2 + d(x, y)2 − 2d(p, x)d(x, y) cos ∠pxy
into the square of (3.2) yieldsed(p, x)2 − 2d(p, x)d(x, y) cos ∠pxy
≥ (1 − 2β)2 d(p, x)2 + 2(1 − 2β)d(x, y)d(p, x).
7

(3.2)

After regrouping we have2d(p, x)(β − β 2 ).
d(x, y)Then it follows from the assumption d(x, y) ≥ d(p, x) sin ϵ and the definition(3.1) of β = β(ϵ) thatecos ∠pxy≤ (2β − 1) +2(β − β 2 )2(1 + sin ϵ)< −1 +
β = − cos ϵ.
sin ϵ
sin ϵ
e
e
Therefore ∠pxy
> π − ϵ and hence ∠xpy
< ϵ. This completes the proof. □ecos ∠pxy≤ (2β − 1) +Let X be a metric space and Γ be some set of minimizing geodesics in X.
We say that Γ is a quasi-bicombing on X if for every x, y ∈ X there is γ ∈ Γwith endpoints x and y (in particular, X is a geodesic space).
Definition 7 (ATB∗ (ϵ)-condition). Let (X, d) be a metric space, Γ be aquasi-bicombing on X and 0 < ϵ < π/2. We say that p ∈ X satisfies the
ATB∗ (ϵ)-condition with respect to Γ if there are L ∈ N and R > 0 suchthat, for every y1 , . . . , yL ∈ BR (p) \ {p} and any γ1 , . . . , γL ∈ Γ where γi isconnecting p and yi , we can find i ̸= j satisfying
min d(yi , x) ≤ β(ϵ)d(p, yi ),x∈γjfor β(ϵ) defined in (3.1).
It follows from Lemma 6 that, if a point p in a geodesic space satisfies
the ATB∗ (ϵ)-condition with respect to some quasi-bicombing for constants
L ∈ N and R > 0, then it also satisfies the ATB(ϵ)-condition with thesame constants. An advantage of the ATB∗ (ϵ)-condition will be seen in
Lemma 8(2).
4. Tools for proving ATB(ϵ)
In this section we provide two lemmas those will be our main ingredientsfor proving the ATB(ϵ)-condition for all the examples we discuss. The firstlemma will be used to study, for example, the lower curvature bounds. The
second lemma will be used in the upper curvature bounds.
Lemma 8. Let (X, dX ) and (Y, dY ) be metric spaces, p ∈ X, q ∈ Y , R > 0,and Φ : BR (q) −→ X be a map with Φ(q) = p.
(1) The simple version. Suppose that Φ satisfies the following conditions:(a) dX (p, Φ(y)) = dY (q, y) for every y ∈ BR (q),(b) dX (Φ(y), Φ(y ′ )) ≥ dY (y, y ′ ) for every y, y ′ ∈ BR (q).
If there are L ∈ N and 0 < ϵ < π/2 such that p satisfies the ATB(ϵ)condition for L and R, then q also satisfies the ATB(ϵ)-condition for thesame L and R.
8

(2) The ATB∗ -version. Suppose that there are quasi-bicombings ΓX and ΓY
on X and Y satisfying the following conditions:(a) dX (p, Φ(y)) ≤ K1 dY (q, y) for some K1 > 0 and every y ∈ BR (q),(b) dX (Φ(y), Φ(y ′ )) ≥ K2 dY (y, y ′ ) for some K2 > 0 and every y, y ′ ∈BR (q),(c) For every γ ∈ ΓY having q as an endpoint, we have Φ(γ) ∈ ΓX .
If there are L ∈ N and 0 < ϵ < π/2 with β(ϵ) < K2 /4K1 such thatp satisfies the ATB∗ (ϵ′ )-condition with respect to ΓX for L and K1 R,
then q satisfies the ATB∗ (ϵ)-condition with respect to ΓY for L, R, and0 < ϵ < π/2 satisfying β(ϵ′ ) ≤ K2 β(ϵ)/K1 .
Proof. (1) Let y1 , . . . , yL ∈ BR (q) \ {q}. Note that Φ(yi ) ̸= p by (1a). Thenthe ATB(ϵ)-condition for p gives i ̸= j such thate∠Φ(yi )pΦ(yj ) < ϵ.
Combining this with the hypotheses (1a) and (1b) implies
e i qyj < ϵ.
∠yThis completes the proof of the simple version.
(2) Let y1 , . . . , yL ∈ BR (q) \ {q} and take γ1 , . . . , γL ∈ ΓY such that γiconnects q and yi . By (2c) we have Φ(γ1 ), . . . , Φ(γL ) ∈ ΓX , and (2a) and(2b) ensure Φ(yi ) ∈ BK1 R (p) \ {p}. The ATB∗ (ϵ′ )-condition for p yields i ̸= jsuch that()()
min dX Φ(yi ), x ≤ β(ϵ′ )dX p, Φ(yi ) .
(4.1)
x∈Φ(γj )Fix x ∈ Φ(γj ) for which this minimum is achieved, and take y ∈ γj such that
Φ(y) = x. By (2b) and (4.1) we have(
) β(ϵ′ ) ()
1dY (yi , y) ≤dX Φ(yi ), x ≤dX p, Φ(yi ) .
K2
K2
Moreover, we deduce from (2a) that) β(ϵ′ )K1β(ϵ′ ) (dX p, Φ(yi ) ≤dY (q, yi ) ≤ β(ϵ)dY (q, yi ).
K2
K2
Thus we havemindY (yi , y ′ ) ≤ β(ϵ)dY (q, yi )
′
y ∈γj

□

and this completes the proof.

Definition 9 (Locally extendable geodesics). Let (X, d) be a geodesic metricspace. We say that p ∈ X satisfies the locally extendable geodesics condition
with a parameter δ > 0 if every unit speed minimizing geodesic γ : [0, T ] −→Bδ (p) can be extended to a unit speed minimizing geodesic γ
e : [0, T + δ] −→X, i.e., γe(t) = γ(t) for all t ∈ [0, T ]. We say that X is a space with locally9extendable geodesics if every p ∈ X satisfies the locally extendable geodesics
condition with some δ(p) > 0.
Definition 10 (Local rough Busemann condition; LRB). Let (X, d) be ageodesic metric space. We say that p ∈ X satisfies the local rough Busemanncondition, the LRB-condition for short, with parameters H, K > 0 if, forevery pair of unit speed minimizing geodesics γ1 : [0, L1 ] −→ BH (p) and
γ2 : [0, L2 ] −→ BH (p) with γ1 (0) = γ2 (0), we have()()
d γ1 (tL1 ), γ2 (tL2 ) ≤ Ktd γ1 (L1 ), γ2 (L2 )for all t ∈ [0, 1].
A subset of a metric space is said to be r-separated if every pair of distinct
points in that set is of distance ≥ r. We will denote by SR (p) the sphere ofcenter x and radius R, namely the set of points y of d(p, y) = R.
Lemma 11. Let X be a geodesic metric space, R > 0, and p ∈ X be a point
satisfying the locally extendable geodesics condition with δ = R and the LRBcondition with H = R and some K > 0. Let 0 < ϵ < π/2 and suppose thatthe cardinality of any (β(ϵ)R/K)-separated set in SR (p) is less than L ∈ N.
Then p satisfies the ATB∗ (ϵ)-condition with respect to any quasi-bicombing
Γ on X for L and R.
We remark that in Lemma 11 everything happens in BR (p), where the
LRB-condition ensures the uniqueness of a minimizing geodesic connecting
any pair of points. Thus nothing really depends on the quasi-bicombing Γ.
Proof. Let y1 , . . . , yL ∈ BR (p) \ {p} and γ1 , . . . , γL be the unique minimizing
geodesics connecting p and these points. We need to find i ̸= j such thatmin d(yi , x) ≤ β(ϵ)d(p, yi ).
x∈γj

(4.2)

By contradiction suppose that (4.2) does not hold for any i ̸= j. For 1 ≤k ≤ L, let y k be the intersection of an extension of γk beyond yk with thesphere SR (p), provided by the locally extendable geodesics condition. Then
we shall see thatβ(ϵ)Rd(y i , y j ) >(4.3)
Kholds for all i ̸= j. Let d(p, yi ) ≤ d(p, yj ) without loss of generality and takex ∈ γj such that d(p, yi ) = d(p, x). We deduce from the negation of (4.2)
that d(yi , x) > β(ϵ)d(p, yi ). Then the LRB-condition implies (4.3) as
R
β(ϵ)Rd(yi , x) >.
d(y i , y j ) ≥Kd(p, yi )KThis contradicts the hypothesis on the cardinality of (β(ϵ)R/K)-separated
sets, and completes the proof.
□105. Examples of ATB(ϵ)-spaces
We present examples of metric spaces satisfying the ATB(ϵ)-condition locally, semi-globally or globally (recall Definition 2 for the definition). The
ATB(ϵ)-condition turns out flexible and covers both Riemannian and Finsler
spaces, as well as both lower and upper curvature bounds.
5.1. Metric spaces with upper curvature bound. We start with metric spaces with upper curvature bounds. We refer to [5] for the basics ofCAT(k)-spaces (geodesic metric spaces of sectional curvature ≤ k). Notethat CAT(0)-spaces are nonpositively curved in the sense of Busemann (Busemann NPC for short).
Proposition 12 (ATB of CAT(k)-spaces). (1) For any k ∈ R and 0 < ϵ <π/2, complete, locally compact CAT(k)-spaces with locally extendable
geodesics are locally ATB(ϵ).
(2) For any 0 < ϵ < π/2, complete, locally compact Busemann NPC spaces
with locally extendable geodesics are semi-globally ATB(ϵ).
Proof. (1) Let (X, d) be a locally compact CAT(k)-space satisfying the locally extendable geodesics condition with δ > 0. Fix x ∈ X. By [27, Section3], there exist 0 < R ≤ δ/2 and K > 0 such that every p ∈ BR (x) satisfies the
LRB-condition for H = R and K. Since X is a complete, locally compact
geodesic space, it is proper and hence B2R (x) is compact (see [5, Proposition 2.5.22]). Thus there exists L ∈ N such that the cardinality of any(β(ϵ)R/K)-separated set in B2R (x) is less than L. Then Lemma 11 showsthat p satisfies the ATB∗ (ϵ)-condition with respect to any quasi-bicombing
Γ on X for L and R. We finally apply Lemma 6 to see that p satisfies the
ATB(ϵ)-condition with the same parameters L and R.
(2) In this case R can be taken arbitrarily large and we have the semiglobal ATB(ϵ) by the same discussion.
□
We present two results as a corollary for later use. For k < 0 and n ∈ N,
we denote by Hn (k) the n-dimensional hyperbolic space. In a normed space,
we call the linear quasi-bicombing the one which consists of all line segments.
Corollary 13. (1) For k < 0, n ∈ N, R > 0 and 0 < ϵ < π/2, there
exists L = L(k, n, R, ϵ) ∈ N such that every point p ∈ Hn (k) satisfies the
ATB(ϵ)-condition for L and R.
(2) For n ∈ N and 0 < ϵ < π/2, there exists L = L(n, ϵ) ∈ N such that every
point p in the Euclidean space Rn satisfies the ATB∗ (ϵ)-condition withrespect to the linear quasi-bicombing for L and R = ∞.
Proof. The first assertion is immediate from Proposition 12(2). The second
assertion follows from Lemma 11 by noticing K = 1 and the fact that the11cardinality of (β(ϵ)R)-separated sets in BR (p) is bounded above independentof R.
□
5.2. Metric spaces with lower curvature bound. Next we consider
finite-dimensional Alexandrov spaces of curvature bounded below. We again
refer to [5] for the basics of those spaces.
Proposition 14 (ATB of Alexandrov spaces). For any k ∈ R and n ∈ N,
every n-dimensional Alexandrov space of curvature ≥ k satisfies the semiglobal ATB(ϵ)-condition for any 0 < ϵ < π/2. Moreover, if k ≥ 0, then it
satisfies the global ATB(ϵ)-condition for any 0 < ϵ < π/2.
Proof. In the first assertion, it suﬃces to consider the case of k < 0. Let(X, d) be an n-dimensional Alexandrov space of curvature ≥ k, and compare
X with the hyperbolic space Hn (k). Fix R > 0 and 0 < ϵ < π/2. Recall
from Corollary 13(1) that there exists L ∈ N such that every point in Hn (k)satisfies the ATB(ϵ)-condition for L and R.
We claim that every point p ∈ X satisfies the ATB(ϵ)-condition for L and
R. If not, then (by the openness of the condition) any point x ∈ Bω (p) with
suﬃciently small ω > 0 does not satisfy the ATB(ϵ)-condition for L and R.
Note that almost every point in Bω (p) has the Euclidean tangent cone, andlet us fix x̄ as one of such points. Now we define the map Φ : BR (x̄) −→ Tx̄ X
as follows, where Tx̄ X is equipped with the hyperbolic metric (thus isometricto Hn (k)). For x ∈ BR (x̄), let γx be a minimizing geodesic from x̄ to x, anddefine Φ(x) := vx ∈ Tx̄ X as the velocity vector of γx with length d(x̄, x). By
the monotonicity of the comparison angle in Alexandrov spaces, Φ satisfies
the conditions of Lemma 8(1). Therefore x̄ satisfies the ATB(ϵ)-condition
for L and R. This is, however, a contradiction and completes the proof fork < 0.
In the case of k = 0, the claim follows from the same discussion and
Corollary 13(2).
□5.3. Finite-dimensional normed spaces. Recall that, in a normed space,
the linear quasi-bicombing means the one which consists of all line segments.
Proposition 15 (ATB of normed spaces). For n ∈ N and 0 < ϵ < π/2,
there exists LN = LN (n, ϵ) such that every point in any n-dimensional
normed space satisfies the ATB∗ (ϵ)-condition with respect to the linear quasibicombing for LN and R = ∞. In particular, every n-dimensional normed
space satisfies the global ATB(ϵ)-condition for LN .
Proof. Let (X, ∥ · ∥) be an n-dimensional normed space. We show the claim
by comparing the norm ∥ · ∥ with the Euclidean one | · |E on X provided by
the John ellipsoid, satisfying ∥x∥ ≤ |x|E ≤ n∥x∥ for all x ∈ X.
12

Notice that the identity map id : (X, ∥ · ∥) −→ (X, | · |E ) satisfies theproperties (2a, 2b, 2c) of Lemma 8(2) for R = ∞, K1 = n, K2 = 1 andthe linear quasi-bicombings. Take ϵ′ > 0 satisfying β(ϵ′ ) ≤ β(ϵ)/n. By
Corollary 13(2), there exists L = L(n, ϵ′ ) such that every point in (X, | · |E )satisfies the ATB∗ (ϵ′ )-condition with respect to the linear quasi-bicombingfor L and R = ∞. Then, by Lemma 8(2), every point in (X, ∥ · ∥) satisfies
the ATB∗ (ϵ)-condition with respect to the linear quasi-bicombing for L andR = ∞.
□
Combining this with Theorem 3 provides an alternative proof of the result in [35], the rectifiability of bounded self-contracted curves in finitedimensional normed spaces.
5.4. Finsler manifolds. From the study of the case of normed spaces in
the previous subsection, it is natural to expect a generalization to Finsler
manifolds. We refer to the books [3, 34] for the basics of Finsler geometry.
By a C ∞ -Finsler manifold we mean a connected C ∞ -manifold M withoutboundary equipped with a nonnegative function F : T M −→ [0, ∞) satisfying:(1) F is C ∞ on T M \ 0,(2) F (cv) = |c|F (v) for all v ∈ T M and c ∈ R,
(3) For any x ∈ M , the function F 2 |Tx M has the positive-definite Hessian on
Tx M \ {0}.
In general we assume the homogeneity F (cv) = cF (v) only for c > 0,
however, in this article we assume the reversibility (F (−v) = F (v)) in orderto restrict ourselves to metric spaces in the ordinary sense for simplicity.
Associated with F one can define the distance function∫ 1()
d(x, y) := inf
F η̇(t) dt,
η0

where η : [0, 1] −→ M runs over all piecewise C 1 -curves from x to y. (Thereversibility implies d(y, x) = d(x, y).) A C ∞ -curve γ : I −→ M froman interval I ⊂ R is called a geodesic if it is locally d-minimizing and hasconstant speed (F (η̇) is constant). We say that (M, F ) is complete if any
geodesic γ : [0, 1] −→ M can be extended to a geodesic from R. In otherwords, the exponential map exp (sending v ∈ T M to γ(1) with γ being thegeodesic with γ̇(0) = v) is well-defined on the whole tangent bundle.
Proposition 16 (Local ATB of Finsler manifolds). Every Finsler manifold(M, F ) is locally ATB(ϵ) for any 0 < ϵ < π/2.
Proof. Fix arbitrary x ∈ M and take small R > 0 such that any two points
in BRM (x) are connected by a unique minimizing geodesic and, for any p ∈
13

T M

BRM (x), the exponential map expp : BRp (0) −→ BRM (p) is C-bi-Lipschitz
for some C ≥ 1 in the sense that()
C −1 F (v ′ − v) ≤ d expp (v), expp (v ′ ) ≤ CF (v ′ − v)
for all v, v ′ ∈ BRp (0). The uniqueness of minimizing geodesics ensures that
BRM (p) has the unique quasi-bicombing.
M
The map exp−1
p : BR (p) −→ Tp M satisfies the conditions in Lemma 8(2)for K1 = 1, K2 = C −1 , and the unique quasi-bicombings. Take ϵ′ >0 satisfying β(ϵ′ ) ≤ β(ϵ)/C. By Proposition 15, 0 ∈ Tp M satisfies the
ATB∗ (ϵ′ )-condition for L = LN (n, ϵ′ ) and R = ∞, where n = dim M . Hence
Lemma 8(2) implies that p satisfies the ATB(ϵ)-condition for L and R. □T M

We can improve Proposition 16 into a global one on Berwald spaces ofnonnegative flag curvature, with the help of an argument in [13]. A Berwald
space is a Finsler manifold whose covariant derivative is independent of the
choice of a reference vector, and then all tangent spaces are mutually linearly
isometric. The flag curvature is a generalization of the sectional curvature in
Riemannian geometry. See [3] or [13] for the definition and more propertiesof Berwald spaces. For example, Riemannian manifolds and (Minkowski)
normed spaces are Berwald spaces.
Proposition 17 (ATB of Berwald spaces). For n ∈ N and 0 < ϵ < π/2,
there exists LB = LB (n, ϵ) such that every complete n-dimensional Berward
space (M, F ) of nonnegative flag curvature is globally ATB(ϵ) for LB .
Proof. Fix p ∈ M , y, z ∈ M \ {p} and let η, ξ : [0, 1] −→ M be minimal
geodesics from p to y, p to z, respectively. If there is no conjugate point top, then being a Berwald space of nonnegative curvature implies()()
d η(t), ξ(t) ≥ td η(1), ξ(1)(5.1)
for all t ∈ [0, 1] (this is the so-called Busemann concavity, see [15, Remark
before Lemma 2.7]). In general, as in [13], we can find a Riemannian metric g
which is aﬃnely equivalent to F , having the nonnegative sectional curvature,and satisfies√
Cn−1 F (v) ≤ g(v, v) ≤ Cn F (v)for all v ∈ T M for some constant Cn ≥ 1 depending only on n (the Binet–
Legendre metric in [24] does this job for example). Since the Busemannconcavity (5.1) globally holds for complete Riemannian manifolds of nonnegative sectional curvature, we obtain
(
)
(
)
(
)
dF η(t), ξ(t) ≥ Cn−1 dg η(t), ξ(t) ≥ Cn−1 tdg η(1), ξ(1)()≥ Cn−2 tdF η(1), ξ(1) .
14

By taking the limit of dF (η(t), ξ(t))/t as t → 0, we have dF (η(1), ξ(1)) ≤˙ − η̇(0)). Hence one can finish the proof by applying Lemma 8(2)Cn2 F (ξ(0)
in the same way as in Proposition 16 with R = ∞.
□5.5. Cayley graphs of virtually abelian groups. In this subsection we
prove the following.
Proposition 18 (ATB of Cayley graphs). Let G be a finitely generated
virtually abelian group with a given symmetric set of generators, and K be
the Cayley graph of G with respect to this set of generators. Then K is
globally ATB(ϵ) for every ϵ > 0.
Being virtually abelian means that G has an abelian subgroup of finite
index. The proof of Proposition 18 is based on the observation that the
Cayley graph of a virtually abelian group is of finite Gromov–Hausdorﬀ
distance from a normed space. This observation follows from Theorem 5
below, which generalizes a result by D. Burago [4] (see also [6, 12]) for theuniversal covering of a torus. The proof mainly follows the same lines.
5.5.1. Periodic metrics.
Theorem 5. Let (X, d) be a proper, geodesic metric space on which Γ = Zn
acts cocompactly and properly discontinuously by isometries. Suppose that
there is a continuous map F : X −→ Rn such that the action Γ ↷ X isequivariant to the standard action Zn ↷ Rn by shifts. Then there exists a
norm ∥ · ∥ of Rn such that the Gromov–Hausdorﬀ distance between (X, d)and (Rn , ∥ · ∥) is finite.
Proof. Fix x0 ∈ X and assume F (x0 ) = 0 without loss of generality. Define
the functiond(x0 , g k (x0 ))∥g∥ := lim.
k→∞kThis limit exists by Fekete’s sub-additive lemma because the function k 7−→
d(x0 , g k (x0 )) is sub-additive, and the limit does not depend on the choice ofx0 . Note that the function ∥ · ∥ satisfies the homogeneity∥g m ∥ = |m| · ∥g∥(5.2)

for m ∈ Z as well as the triangle inequality∥g · h∥ ≤ ∥g∥ + ∥h∥(5.3)

for g, h ∈ Γ (see, e.g., [19]).
We need two more lemmas to finish the proof of the theorem. The first one
is an easy observation on sub-additive functions and the proof is omitted.
15

Lemma 19. Let f : Z −→ (0, ∞) be a sub-additive function such thatlimn→∞ f (n) = ∞ and f (n) ≤ f (2n)/2 + C for some C > 0 and all n ∈ Z.
Then we have c := limn→∞ f (n)/n > 0 and cn ≤ f (n) ≤ cn + 2C for alln ∈ Z.
The next lemma is the main technical ingredient of the proof of Theorem 5,that allows us to apply Lemma 19 to the function d(x0 , g k (x0 )).
Lemma 20. There exists C ≥ 0 such that d(x0 , g(x0 )) ≤ d(x0 , g 2 (x0 ))/2 + C
holds for every g ∈ Γ.
Let us postpone the proof of Lemma 20 and finish the proof of the theorem.
It follows from Lemmas 19 and 20 that ∥ · ∥ is bounded away from 0. Hence,
together with (5.2) and (5.3), we find that ∥·∥ defines a norm of Rn including
Zn = Γ. It also follows from Lemma 19 that (Γ(x0 ), d) is of finite Gromov–
Hausdorﬀ distance from (Γ, ∥·∥). Since Γ acts cocompactly on X, Theorem 5follows.
□
It remains to prove Lemma 20.
Proof of Lemma 20. The proof is based on the following analogue of theintermediate value theorem proven in [4].
Lemma 21. Let f : [0, 1] −→ Rn be a continuous map with f (1) = 0. Then
there exist 0 ≤ a1 ≤ a2 ≤ · · · ≤ an ≤ 1 such that
n
∑
f (0)
(−1)i+1 f (ai ) =.
2
i=1
In other words, if we set a0 = 0 and an+1 = 1, then we have
∑ (
∑ (
)
) f (0)f (ai ) − f (ai+1 ) =f (ai ) − f (ai+1 ) =.
20≤i≤n,
0≤i≤n,i≡1 mod 2(5.4)

i≡0 mod 2

The next lemma is straightforward.
Lemma 22. For any r > 0 there exists R ≥ 0 such that, for every x1 , x2 ∈ X
with |F (x1 ) − F (x2 )| ≤ r, we have d(x1 , x2 ) ≤ R. We will denote theminimum of such R by Θ(r).
Fix g ∈ Γ. The plan of the proof is as follows. We will find a path
Π : [0, 1] −→ X connecting g 2 (x0 ) and x0 satisfying the following.
(1) Π is an almost shortest path in the sense that()√
L(Π) ≤ d x0 , g 2 (x0 ) + 2nΘ( n).
(5.5)(2) Π is almost containing two copies of a shortest path between g(x0 )and x0 in a certain sense, and we have()√L(Π) ≥ 2d x0 , g(x0 ) − 2Θ(n n).
(5.6)
16

Notice that Lemma 20 readily follows from (5.5) and (5.6).
Step 1 : The construction of Π. Let γ : [0, 1] −→ X be a minimizing
geodesic from g 2 (x0 ) to x0 . We apply Lemma 21 to the map γ
e = F ◦γ(recall F (x0 ) = 0) and obtain 0 = a0 ≤ a1 ≤ a2 ≤ · · · ≤ an ≤ an+1 = 1 such
that
∑

(

)
γ
e(ai ) − γe(ai+1 ) =∑(0≤i≤n,
i≡0 mod 20≤i≤n,i≡1 mod 2

) γ
e(0)
γ
e(ai ) − γe(ai+1 ) =
.
2

(5.7)

Let [·] : Rn −→ Zn be the coordinate floor function, namely [x]j ≤ xj <[x]j +1. We fix some pi ∈ F −1 ([eγ (ai )]) for 1 ≤ i ≤ n, and also set p0 := g 2 (x0 )and pn+1 := x0 . Now we define Π : [0, 1] −→ X as the concatenation of theshortest paths pi pi+1 , where Π(ai ) = pi .
√Step 2 : The proof of (5.5). We
γ (ai )] − γe(ai )| ≤ n and√ deduce from |[eLemma 22 that d(pi , γ(ai )) ≤ Θ( n). Then the triangle inequality implies(5.5) asL(Π) =

n
∑

n
∑
(
)
√
d(pi , pi+1 ) ≤d γ(ai ), γ(ai+1 ) + 2nΘ( n)

i=0

i=0

(
)
√
= d x0 , g 2 (x0 ) + 2nΘ( n),where we have 2n instead of 2(n + 1) since p0 = g 2 (x0 ) = γ(0) and pn+1 =x0 = γ(1).
Step 3 : The proof of (5.6). For 0 ≤ i ≤ n, we denote by gi an element ofΓ such that gi (pi+1 ) = pi . Then we can write
L(Π) =

n
∑

n
∑
(
)
d(pi , pi+1 ) =d gi (pi+1 ), pi+1 .

i=0

i=0

Notice that, if x and y are in the same orbit of an isometric action of anabelian group, then for any element α of this group we have d(x, α(x)) =d(y, α(y)). Thus we observe
∑
∑
)
(
)
(
d gi (pi+1 ), pi+1
L(Π) =
d gi (pi+1 ), pi+1 +0≤i≤n,
i≡0 mod 2

∑

=

( ((
d gi

0≤i≤n,
i≡0 mod 2+∑0≤i≤n,i≡1 mod 20≤i≤n,i≡1 mod 2)) (gk (x0 ) ,∏i<k≤n,
k≡0 mod 2

( ((
d gi

∏

i<k≤n,
k≡1 mod 2

17∏)
gk (x0 )i<k≤n,
k≡0 mod 2)) (gk (x0 ) ,∏i<k≤n,
k≡1 mod 2))
gk (x0 ) .
)Applying the triangle inequality to each of those sums provides
(( ∏
)
(( ∏
)
)
)
gk (x0 ), x0 + d
gk (x0 ), x0 .
L(Π) ≥ d0≤k≤n,
k≡0 mod 20≤k≤n,
k≡1 mod 2

Now, again by the triangle inequality, to provide (5.6) it suﬃces to show thatfor m = 1, 2
(( ∏
)
)
√
d
gk (x0 ), g(x0 ) ≤ Θ(n n).
0≤k≤n,k≡m mod 2

Thanks to Lemma 22, the above inequality follows from
(( ∏
)
)
(
)
√
F
gk (x0 ) − F g(x0 ) ≤ n n.

(5.8)

0≤k≤n,k≡m mod 2

Let gek ∈ Zn (resp. ge) the element corresponding to gk (resp. g). Then, since
F is Γ-equivariant, we have
(( ∏
)
)
∑
(
)
F
gk (x0 ) − F g(x0 ) =gek − ge.
0≤k≤n,k≡m mod 20≤k≤n,k≡m mod 2

Therefore (5.8) is rewritten as
∑ (
) γ
√
e(0)
[e
γ (ak )] − [eγ (ak+1 )] −≤ n n,
20≤k≤n,k≡m mod 2√
which follows from (5.7), |[eγ (ai )] − γe(ai )| ≤ n for 1 ≤ i ≤ n and γe(ai ) =[eγ (ai )] for i = 0, n + 1. This completes the proof of Lemma 20 and thenTheorem 5.
□
Applying Theorem 5 to the setting of Proposition 18 yields the following.
Corollary 23. Let G be a finitely generated virtually abelian group with agiven symmetric set of generators and (K, ρ) be the Cayley graph of G withrespect to this set of generators. Then there exists a norm ∥ · ∥ on Rn suchthat the Gromov–Hausdorﬀ distance between (K, ρ) and (Rn , ∥ · ∥) is finite.
Proof. Let Γ ∼ Zn be a subgroup of finite index in G (we have n < ∞because any subgroup of finite index in a finitely generated group is finitely
generated). Then Γ acts on K cocompactly and properly discontinuously.
Let us construct a map F : K −→ Rn satisfying the conditions of Theorem 5(without using the Bieberbach theorem). Let x0 ∈ K correspond to e ∈ G.
We fix an isomorphism g 7−→ ge between Γ and Zn and map the orbit Γ(x0 ) to
the integer lattice via this isomorphism. Then choose one representative fromevery right coset with respect to Γ, say g1 , . . . , gk , and map g1 (x0 ), . . . , gk (x0 )18

to Rn in an arbitrary way. Then define the map on G(x0 ) by F (hgi (x0 )) :=F (gi (x0 )) + e
h for h ∈ Γ. Finally extend F linearly to the edges. Then
we apply Theorem 5 to obtain that K is of finite distance from some ndimensional normed space.
□
5.5.2. Proof of Proposition 18. For a, b, c > 0 satisfying the triangle inequalitiesa ≤ b + c,b ≤ c + a,c ≤ a + b,
(5.9)
let us denote by ∠(a, b; c) the angle between a and b for the flat triangle
with edges a, b, c, namely c2 = a2 + b2 − 2ab cos ∠(a, b; c). By an elementary
calculation we observe the following.
Lemma 24. For every d, δ > 0, there exists D = D(d, δ) > 0 such that,for any triples a, b, c > 0 and a′ , b′ , c′ > 0 satisfying (5.9), a, b > D, c > 0,|a − a′ | ≤ d, |b − b′ | ≤ d and |c − c′ | ≤ d, we have |∠(a′ , b′ ; c′ ) − ∠(a, b; c)| ≤ δ.
The next lemma is crucial in the proof of Proposition 18.
Lemma 25. Let (X, dX ) be a metric space and ϵ > 0 such that, for every
R > 0, there is LX = LX (R) > 0 for which every point p ∈ X satisfies ATB(ϵ) with constants LX and R. Suppose further that X is of finite
Gromov–Hausdorﬀ distance from some metric space (Y, dY ) which is globally
ATB(ϵ/2). Then X is globally ATB(ϵ).
Proof. Denote the Gromov–Hausdorﬀ distance between X and Y by d/2, andtake D = D(d, ϵ/2) from Lemma 24. Let then LX = LX (D) be the constantprovided by the hypothesis on X, and Y be globally ATB(ϵ/2) with LY . We
shall show that X is globally ATB(ϵ) with the constant LX + LY − 1. To thise i pyj ≥ ϵ for all i ̸= j.
end, given p ∈ X, let y1 , y2 , . . . , yN ∈ X \ {p} satisfy ∠yThen, on the one hand, we immediately find #[{y1 , y2 , . . . , yN } ∩ BD (p)] <LX . On the other hand, we obtain #[{y1 , y2 , . . . , yN } \ BD (p)] < LY since,otherwise, the choice of d and Lemma 24 provide a point p′ ∈ Y and LY
e ′ p′ y ′ ≥ ϵ/2 for all i ̸= j and we have apoints yi′ ∈ Y \ {p′ } such that ∠yijcontradiction with the choice of LY . Therefore N ≤ LX + LY − 2 and thiscompletes the proof.
□
Now we are ready to prove Proposition 18. Note that the Cayley graph K
satisfies the first condition of Lemma 25. Then Corollary 23 and Lemma 15
ensure that Lemma 25 is indeed available with Y a finite-dimensional normed
space, and we obtain Proposition 18.
5.6. Submetries and Quotient spaces. We finally discuss the behaviorof the ATB-condition under some deformations of metric spaces, related toLemma 8.
19

Definition 26 (Submetries). Let (X, dX ) and (Y, dY ) be metric spaces. A
map ψ : X −→ Y is called a submetry if, for every x ∈ X and r > 0, theimage of the closed ball Br (x) coincides with Br (ψ(x)).
Notice that every submetry is 1-Lipschitz.
Proposition 27. Let ψ : X −→ Y be a submetry and 0 < ϵ < π/2. Supposethat p ∈ X satisfies the ATB(ϵ)-condition for some L ∈ N and R > 0. Thenq := ψ(p) ∈ Y also satisfies the ATB(ϵ)-condition for L and R.
Proof. It follows from the definition of the submetry that, for every y ∈ Y ,we have SdY (q,y) (p) ∩ ψ −1 (y) ̸= ∅. Thus we have (by the axiom of choice)Φ : Y −→ X such that Φ(y) ∈ SdY (q,y) (p) ∩ ψ −1 (y) and Φ(q) = p. Sincey = ψ(Φ(y)) and ψ is 1-Lipschitz, we conclude that Φ satisfies the conditionsof Lemma 8(1). This completes the proof.
□
For a metric space (X, dX ) with an isometric group action G ↷ X withclosed orbits, the quotient metric space (X/G, dX/G ) is defined in the following way. Denoted by X/G is the set of orbits, and the metric is given
bydX/G (x, y) := inf dX (x, y).
x∈x, y∈y

Clearly, the quotient map π : X −→ X/G is a submetry. Thus, we have thefollowing corollary of Proposition 27.
Corollary 28 (ATB of Quotient spaces). Let (X, dX ) satisfy the local, semiglobal, or global ATB(ϵ)-condition and G ↷ X be an isometric group action
with closed orbits. Then X/G also satisfies the local, semi-global, or global
ATB(ϵ)-condition, respectively.
6. Gradient curves of quasi-convex functions are
self-contracted
As we mentioned in the introduction, self-contracted curves arise as gradient curves of quasi-convex functions.
Definition 29 (Quasi-convexity). A function f : X −→ R on a geodesic
metric space (X, d) is said to be quasi-convex if, for any x, y ∈ X and any
minimizing geodesic γ : [0, 1] −→ X from x to y, we have for all t ∈ (0, 1)()
f γ(t) ≤ max{f (x), f (y)}.
An equivalent definition is that every sub-level set {x ∈ X | f (x) ≤ a}is convex. It was known that gradient curves of quasi-convex functions are
self-contracted in• Euclidean spaces (see [10, Proposition 6.2]),• CAT(0)-spaces (see [30, Proposition 4.6]).
20

In the latter case, to be precise, we assumed an auxiliary condition that f iseither bounded below or λ-convex for some λ ∈ R in the sense that()λ
f γ(t) ≤ (1 − t)f (x) + tf (y) − (1 − t)td(x, y)22
for all t ∈ (0, 1) along every geodesic γ : [0, 1] −→ X from x to y.
In this section we shall add spaces with lower curvature bound into theabove list. Our proof is based on the evolution variational inequality (in thesame spirit as [30]) and applies to other cases where the evolution variational
inequality is known. We remark that normed spaces and Finsler manifolds
do not belong to this class and the self-contractedness may fail (see [31] forthe failure of the contraction property d(γ(t), η(t)) ≤ e−λt d(γ(0), η(0)) for
pairs of gradient curves).
Given a function f : X −→ R on a metric space (X, d), an absolutely continuous curve γ : [0, ∞) −→ X is said to satisfy the λ-evolution variational
inequality (EVIλ for short) if, for any y ∈ X and almost every t > 0,
[
]
)2
(
)2
(
)
d 1 (
d γ(t), y+ λd γ(t), y + f γ(t) ≤ f (y).
(6.1)
dt 2
The existence of curves satisfying the EVIλ characterizes the λ-convexity off , and then those curves turn out gradient curves of f in the metric sense(called the energy dissipation) that means∫()() 1 t{()2 }f γ(t) = f γ(s) −|γ̇(r)|2 + |∇f | γ(r)dr
2 s
for all s < t, where |γ̇| is the metric speed of γ and |∇f | is the (descending)slope of f . The inequality (6.1) possesses an overwhelming importance ingradient flow theory on “Riemannian-like” spaces, we refer to the book [1]for more on gradient flow theory on metric spaces.
Although our main interest is in metric spaces with lower sectional curvature bounds, here we discuss more general Ricci curvature bounds. The
Riemannian curvature-dimension condition RCD(K, ∞) is a synthetic notion, for metric measure spaces, of the lower Ricci curvature bound defined as the combination of the K-convexity of the relative entropy on
the L2 -Wasserstein space and the linearity of the corresponding heat flow(see [2,22,36]). It is known by Sturm [38] that λ-convex functions on locallycompact RCD(K, ∞)-spaces satisfy EVIλ (6.1). This is the starting point ofour proof of the self-contractedness.
In [32] (see also [39]) it was shown that any n-dimensional Alexandrov
space of curvature ≥ k equipped with the n-dimensional Hausdorﬀ measuresatisfies RCD(K, ∞) for K = (n − 1)k. Therefore the following proposition
covers the situation treated in Corollaries 3, 5. We also refer to [23, 29, 33]
21

for direct studies of gradient curves in Alexandrov spaces as well as in the
Wasserstein spaces over them.
Proposition 30 (Quasi-convexity of gradient curves). Let (X, d, m) be a
locally compact metric space equipped with a Borel measure m, and assume
that it satisfies RCD(K, ∞) for some K ∈ R. Then, for any lower semicontinuous, quasi-convex function f : X −→ R which is also λ-convex forsome λ ∈ R, every gradient curve of f is self-contracted.
Proof. Let γ : [0, ℓ) −→ X be a gradient curve of f and take T ∈ (0, ℓ). Fix
arbitrary t0 ∈ (0, T ) and let ξ : [0, 1] −→ X be a minimal geodesic fromγ(t0 ) to γ(T ) (the existence follows from RCD(K, ∞)). For each s ∈ (0, 1),
we observe from the triangle inequality thatd(γ(t), γ(T ))2 − d(γ(t0 ), γ(T ))22(t − t0 )t→t0()
d(γ(t), ξ(s)) − d(γ(t0 ), ξ(s))≤ d γ(t0 ), γ(T ) lim supt − t0t→t0d(γ(t), ξ(s))2 − d(γ(t0 ), ξ(s))2d(γ(t0 ), γ(T ))lim sup.
=d(γ(t0 ), ξ(s)) t→t0
2(t − t0 )lim supThanks to (the integration of) (6.1) with y = ξ(s), we have()2d(γ(t), ξ(s))2 − d(γ(t0 ), ξ(s))2
lim sup+ λd γ(t0 ), ξ(s)2(t − t0 )
t→t0
(
)
(
)
≤ f ξ(s) − f γ(t0 ) ≤ 0,
where the second inequality follows from the quasi-convexity of f sincef (γ(T )) ≤ f (γ(t0 )) holds for γ being a gradient curve. Letting s → 0,
we obtaind(γ(t), γ(T ))2 − d(γ(t0 ), γ(T ))2
lim sup≤02(t − t0 )t→t0
for all t0 ∈ (0, T ). Hence d(γ(t), γ(T )) is non-increasing in t ∈ [0, T ].

□Let us finally remark that the self-contractedness of gradient curves does
not characterize quasi-convexity, consider for instance f (x) = sin x on R. We
need a “global” property, such as the EVI (between a curve and a point) or
the contraction property (between two curves), for such a characterizationof the global convexity.
In a diﬀerent approach, nevertheless, it was shown in a recent paper[11] that a C 1,α -curve (α ∈ (1/2, 1]) in Rn satisfies a strict version of theself-contractedenss (called the strong self-contractedness) if and only if its
reparametrization is realized as a gradient curve of a C 1 -convex function.
22

7. Doubling condition and absence of large SRA(α)-subsets
In this section we show that the absence of large SRA(α)-subsets implies
the doubling condition. This could be regarded as a weak converse to Theorem 2, and gives another interesting connection between the rectifiabilityof bounded self-contracted curves and the finiteness of dimension. A metricspace (X, d) is said to satisfy the (metric) doubling condition if there exists
L ∈ N such that, for any x ∈ X and R > 0, we can find x1 , . . . , xL ∈ BR (x)∪satisfying BR (x) ⊂ Li=1 BR/2 (xi ).
Theorem 6. Let (X, d) be a metric space, and suppose that there exist
0 < α < 1 and N ∈ N such that X does not admit any SRA(α)-subset of
cardinality ≥ N . Then X satisfies the doubling condition with L = L(N, α).
e ∈ N be such that α(Ne − 2) ≥ 3, and we consider the Ramsey
Proof. Let N
e, N
e, N
e, N
e, N
e, N
e , N ; 3) ∈ N. This means that, if we takenumber H = R(Nan H-element set M, then for every coloring of 3-point subsets of M in
7 colors, we have the following property: Either there exists an N -element
subset of M such that all of its 3-point subsets are of the seventh color, ore -element subset of M such that all of its 3-point subsets
there exists an N
are of the same color from the first to the sixth.
We shall show that the doubling condition holds with L = H − 1. Assume
in contrary that there exist x ∈ X and R > 0 such that the open (R/2)neighborhood of any L-point set in BR (x) cannot cover BR (x). Then one
can inductively choose x1 , . . . , xL+1 ∈ BR (x) with d(xi , xj ) ≥ R/2 for everyi ̸= j. We color triples of points in this set in 7 colors. For i < j < k wecolor (xi , xj , xk ) by the following rule:(1) d(xi , xk ) ≥ d(xi , xj ) + αd(xj , xk ),(2) not (1) but d(xj , xk ) ≥ d(xi , xj ) + αd(xi , xk ),(3) not (1)–(2) but d(xj , xk ) ≥ d(xi , xk ) + αd(xi , xj ),(4) not (1)–(3) but d(xi , xk ) ≥ d(xj , xk ) + αd(xi , xj ),(5) not (1)–(4) but d(xi , xj ) ≥ d(xj , xk ) + αd(xi , xk ),(6) not (1)–(5) but d(xi , xj ) ≥ d(xi , xk ) + αd(xj , xk ),(7) none of the above.
By the definition of L + 1 = H, either there exists an N -point set satisfying
e -point set satisfying one of the properties (1)–(6). The(7) or there exists an N
first case contradicts the assumption on the absence of an N -point SRA(α)set. Now we are going to deal with the second case. By renumbering, lete points in question.
x1 , . . . , xNe be the N
23

Case 1: x1 , . . . , xNe are of the first color. Taking the sum of the inequalitiesd(x1 , x3 ) ≥ d(x1 , x2 ) + αd(x2 , x3 ),d(x1 , x4 ) ≥ d(x1 , x3 ) + αd(x3 , x4 ),...,d(x1 , xNe ) ≥ d(x1 , xNe −1 ) + αd(xNe −1 , xNe ),e,
we have, together with the choice of N()
d(x1 , xNe ) ≥ d(x1 , x2 ) + α d(x2 , x3 ) + · · · + d(xNe −1 , xNe )R
e − 2) R ≥ 2R.
≥ + α(N22This contradicts the assumption that x1 , xNe ∈ BR (x).
Case 2: x1 , . . . , xNe are of the second color. We have the inequalitiesd(x2 , x3 ) ≥ d(x1 , x2 ) + αd(x1 , x3 ),d(x3 , x4 ) ≥ d(x2 , x3 ) + αd(x2 , x4 ),...,d(xNe −1 , xNe ) ≥ d(xNe −2 , xNe −1 ) + αd(xNe −2 , xNe ).
Summing them provides, similarly to the previous case,)(d(xNe −1 , xNe ) ≥ d(x1 , x2 ) + α d(x1 , x3 ) + · · · + d(xNe −2 , xNe ) ≥ 2R,
which is a contradiction.
Case 3: x1 , . . . , xNe are of the third color. In this case we sumd(x2 , xNe ) ≥ d(x1 , xNe ) + αd(x1 , x2 ),d(x3 , xNe ) ≥ d(x2 , xNe ) + αd(x2 , x3 ),...,d(xNe −1 , xNe ) ≥ d(xNe −2 , xNe ) + αd(xNe −2 , xNe −1 )to obtain)(d(xNe −1 , xNe ) ≥ d(x1 , xNe ) + α d(x1 , x2 ) + · · · + d(xNe −2 , xNe −1 ) ≥ 2R.

This is again a contradiction. The cases produced by the forth, fifth and sixth
inequalities can be reduced to the first, second and third cases, respectively,by reversing the order of x1 , . . . , xNe . This completes the proof.
□
8. Counter-examples
This section is devoted to several kinds of examples of metric spaces including infinite (or arbitrarily large) SRA(α)-subsets or unrectifiable bounded
self-contracted curves. Some of them satisfy the doubling condition, therefore the converse of Theorem 6 does not hold in general.
248.1. Heisenberg group. As a first example of particular interest, let us
consider the Heisenberg group (R3 , H) (we refer to [26]). Let γ : [0, 1] −→ R3be√defined by γ(t) = (0, 0, t). The distance between (0, 0, s) and (0, 0, t) is
2 π|s − t|, thus the z-axis is isometric to the (1/2)-snowflake of R. Henceγ is bounded and self-contracted, whereas every small piece of γ has infinite
length.
On the one hand, the Heisenberg group is known to satisfy the measurecontraction property (in the sense of [28,37]) and, in particular, the doubling
condition. On the other hand, it does not satisfy the curvature-dimension
condition CD(K, N ) for any parameters K, N (see [14]).
8.2. Metric trees. For x, y ∈ R2 , we denote by [x, y] ⊂ R2 the segment
connecting x and y. Let {ti }∞i=1 ⊂ (0, 1] be a strictly decreasing sequencesuch that limi→∞ ti = 0, and define B ⊂ R2 by∞∪
B = [(0, 0), (1, 0)] ∪ [(ti , 0), (ti , ti )].
i=1

We consider B as a metric space with the intrinsic distance denoted by dB .
Note that (B, dB ) is a compact metric tree, thus it is CAT(k) for any k ∈ R,
whereas the local extendability of geodesics fails. We put yi := (ti , ti ) andE := {yi }∞i=1 . For i > j, we havedB (yi , yj ) = 2tj .
Thus for i > j > k we have
2tk = dB (yi , yk ) = dB (yj , yk ) ≥ dB (yi , yj ) = 2tj .
Therefore the infinite set E satisfies the SRA(α)-condition for every α > 0(moreover, it is a ultrametric space). Compare this with Proposition 12.
−i+1
Example 31. Consider {ti }∞. In this case (B, dB )i=1 given by ti = 2satisfies the doubling condition. Note also that the inclusion id : B −→ R2
provides a bi-Lipschitz embedding into the Euclidean plane (then what is
missed from Lemma 8(2) is the last condition (2c)).
−1
Example 32. Take {ti }∞i=1 given by ti = i . Then E provides a (discrete)bounded self-contracted curve of the infinite length.

8.3. Laakso graphs. Next we are going to find SRA(α)-subsets in Laakso
graphs. Let us give the definition and basic properties of Laakso graphs
following [25] (see also [16, 17]). The family of Laakso graphs {GN }∞N =0 isdefined via induction as in Figure 1. Precisely, we start with G0 consisting of
one edge of unit length, and GN is constructed by six copies of GN −1 whose
metrics are scaled by a factor of 1/4. Glue four of these copies cyclicallyby identifying pairs of endpoints, and attach at two opposite gluing points
25

G0
r

G1
r

G2
r

Figure 1. Laakso graphs.
the remaining two copies. Then GN is the resulting graph equipped with
the intrinsic distance (thus the diameter of GN is again 1). Laakso graphs
satisfy the doubling condition and local Poincaré inequality.
Proposition 33. For any n, N ∈ N with n ≤ N , there exists an n-point
SRA(3/5)-subset X in GN .
The construction of X will be given by Example 34 below. Before that, we
explain the necessary notions. Fix one of the two vertexes of GN of degreeone. We call it the root of GN and denote by r (see Figure 1). Note thatin no edge the two endpoints are at the same distance from the root. Thus
we can direct all the vertexes from the endpoint which is closer to r to theendpoint further from r. It is easy to see that any vertex has at most twooutgoing edges with respect to this direction structure. For a vertex which
has two outgoing edges, we say that one of them goes to the right and theother one goes to the left. We make such a choice in an arbitrary way foreach vertex.
Example 34. Fix n, N ∈ N such that n ≤ N . Let γ : [0, 1] −→ GN be theunit speed minimizing geodesic from r to the other end of GN , never goingto the right in the sense above. For each 1 ≤ i ≤ n, we put
(
)
1
1
1
yi := γ++ ··· + i .
4 16
4
26

Then let xi ∈ GN be the endpoint of the oriented path which starts fromyi , never goes to the left and has length 4−i . We define X := {x1 , . . . , xn }.
Note that every xi is a vertex of GN by n ≤ N .
Proof of Proposition 33. We shall show that X in Example 34 satisfies theSRA(3/5)-condition. For 1 ≤ i < k ≤ n, we observe by constructiond(xi , xk ) = d(xi , yi ) + d(yi , yk ) + d(yk , xk )
(
)
1
1
1
1
= i+
+
·
·
·
+
+
.
4
4i+1
4k
4k
Hence we have, for i < j < k,d(xj , xk ) < d(xi , xk ) < d(xi , xj ).
Thus, to show the SRA(3/5)-condition, it is suﬃcient to see
3d(xi , xj ) ≤ d(xi , xk ) + d(xj , xk ).
5This is done by
(
)
1
1
1
1
1
1
3
d(xi , xj ) − d(xi , xk ) = j −+ · · · + k − k < j − j+1 = j+1 ,
j+1
4
4
4
4
4
4
4
and
d(xj , xk ) ≥1
1
5+ j+1 = j+1 .
j
4
4
4

□

9. Questions
Recall from Proposition 30 that gradient curves of lower semi-continuous,quasi-convex, λ-convex functions in locally compact RCD(K, ∞)-spaces are
self-contracted. However, it is not known whether bounded self-contracted
curves are rectifiable in RCD(K, ∞)-spaces. By Theorem 3, to provide anaﬃrmative answer to this question, it suﬃces to show that RCD(K, ∞)spaces are locally ATB(ϵ) for some 0 < ϵ < π/6. It may be in fact morereasonable to consider RCD(K, N )-spaces with N ∈ (1, ∞), since the finitedimensionality is a natural condition for the rectifiability. Thus we have thefollowing question.
Question 35. Do RCD(K, N )-spaces satisfy the local ATB(ϵ)-condition?
Notice that the weaker notion of measure contraction property does not
imply ATB(ϵ) as is evident in the Heisenberg group (recall Subsection 8.1).
Related to Proposition 14, one can also ask the following question. See [7,7.19] for the notion of the boundary of an Alexandrov space of curvature≥ 0.
27

Question 36. Do the boundaries of finite-dimensional Alexandrov spaces ofcurvature ≥ 0 satisfy the local ATB(ϵ)-condition?
Acknowledgements. NL was partially supported by RFBR grant 17-0100128. SO was supported in part by JSPS Grant-in-Aid for Scientific Research (KAKENHI) 15K04844. Theorem 6, Propositions 12, 15, 16 and 27
are supported by the Russian Science Foundation under grant 16-11-10039.
We would like to thank Sergei Ivanov for useful discussions.

References[1] L. Ambrosio, N. Gigli, and G. Savaré. Gradient flows in metric spaces and in thespace of probability measures. Birkhäuser Verlag, 2005.
[2] L. Ambrosio, N. Gigli, and G. Savaré. Metric measure spaces with Riemannian Ricci
curvature bounded from below. Duke Math. J., 163(7):1405–1490, 2014.
[3] D. Bao, S.-S. Chern, and Z. Shen. An introduction to Riemann–Finsler geometry.
Springer-Verlag, 2000.
[4] D. Burago. Periodic metrics. Adv. Soviet Math., 9:205–210, 1992.
[5] D. Burago, Y. Burago, and S. Ivanov. A Course in Metric Geometry, volume 33.
American Mathematical Society, 2001.
[6] D. Burago, S. Ivanov, and B. Kleiner. On the structure of the stable norm of periodic
metrics. Mathematical Research Letters, 4(6):791–808, 1997.
[7] Yu. Burago, M. Gromov, and G. Perel’man. A.D. Alexandrov spaces with curvaturebounded below. Russian Mathematical Surveys, 47(2):1, 1992.
[8] A. Daniilidis, G. David, E. Durand-Cartagena, and A. Lemenant. Rectifiability of selfcontracted curves in the Euclidean space and applications. The Journal of GeometricAnalysis, 25(2):1211–1239, Apr 2015.
[9] A. Daniilidis, R. Deville, E. Durand-Cartagena, and L. Riﬀord. Self-contracted curves
in Riemannian manifolds. J. Math. Anal. Appl., 457:1333–1352, 2018.
[10] A. Daniilidis, O. Ley, and S. Sabourau. Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions. Journal de Mathématiques Pures
et Appliquées, 94(2):183–199, 2010.
[11] E. Durand-Cartagena and A. Lemenant. Self-contracted curves are gradient flows ofconvex functions. Proc. Amer. Math. Soc., to appear.
[12] S. Ivanov. A generalization of intermediate value theorem on Rk . MathOverflow.
URL: https://mathoverflow.net/q/119092 (version: 2013-01-16).
[13] S. Ivanov and A. Lytchak. Rigidity of Busemann convex Finsler metrics. ArXiv eprints, November 2017.
[14] N. Juillet. Geometric inequalities and generalized Ricci bounds in the Heisenberg
group. Int. Math. Res. Not. IMRN, 2009(13):2347–2373, 2009.
[15] M. Kell. Sectional curvature-type conditions on metric spaces. J. Geom. Anal., toappear, 2018.
[16] T. J. Laakso. Plane with A∞ -weighted metric not bilipschitz embeddable to Rn .
Bulletin of the London Mathematical Society, 34(6):667–676, 2002.
[17] U. Lang and C. Plaut. Bilipschitz embeddings of metric spaces into space forms.
Geometriae Dedicata, 87(1-3):285–307, 2001.
28

[18] E. Le Donne, T. Rajala, and E. Walsberg. Isometric embeddings of snowflakes into
finite-dimensional Banach spaces. Proceedings of the American Mathematical Society,146(2):685–693, 2018.
[19] N. Lebedeva. On the fundamental group of a compact space without conjugate points.
Preprint, 2002.
[20] A. Lemenant. Rectifiability of non Euclidean planar self-contracted curves. Confluentes Mathematici, 2017.
[21] M. Longinetti, P. Manselli, and A. Venturi. On steepest descent curves for quasi
convex families in Rn . Mathematische Nachrichten, 288(4):420–442, 2015.
[22] J. Lott and C. Villani. Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2), 169:903–991, 2009.
[23] A. Lytchak. Open map theorem for metric spaces. St. Petersberg. Math. J., 17:477–
491, 2006.
[24] V. S. Matveev and M. Troyanov. The Binet–Legendre metric in Finsler geometry.
Geom. Topol., 16(4):2135–2170, 2012.
[25] M. Mendel and A. Naor. Markov convexity and local rigidity of distorted metrics.
In Proceedings of the twenty-fourth annual symposium on Computational Geometry,pages 49–58. ACM, 2008.
[26] R. Montgomery. A tour of subriemannian geometries, their geodesics and applications. American Mathematical Society, 2002.
[27] S. Ohta. Convexities of metric spaces. Geometriae Dedicata, 125(1):225–250, Mar
2007.
[28] S. Ohta. On the measure contraction property of metric measure spaces. Comment.
Math. Helv., 82:805–828, 2007.
[29] S. Ohta. Gradient flows on Wasserstein spaces over compact Alexandrov spaces.
Amer. J. Math., 131:475–516, 2009.
[30] S. Ohta. Self-contracted curves in CAT(0)-spaces and their rectifiability. J. Geom.
Anal., to appear, 2019.
[31] S. Ohta and K.-T. Sturm. Non-contraction of heat flow on Minkowski spaces. Arch.
Ration. Mech. Anal., 204:917–944, 2012.
[32] A. Petrunin. Alexandrov meets Lott–Villani–Sturm. Münster J. Math., 4:53–64, 2011.
[33] G. Savaré. Gradient flows and diﬀusion semigroups in metric spaces under lower
curvature bounds. C. R. Math. Acad. Sci. Paris, 345:151–154, 2007.
[34] Z. Shen. Lectures on Finsler geometry. World Scientific, 2001.
[35] E. Stepanov and Y. Teplitskaya. Self-contracted curves have finite length. Journal of
the London Mathematical Society, 96(2):455–481, 2017.
[36] K.-T. Sturm. On the geometry of metric measure spaces. I. Acta Math., 196:65–131,
2006.
[37] K.-T. Sturm. On the geometry of metric measure spaces. II. Acta Math., 196:133–177,
2006.
[38] K.-T. Sturm. Gradient flows for semiconvex functions on metric measure spaces existence, uniqueness, and Lipschitz continuity. Proc. Amer. Math. Soc., 146(9):3985–
3994, 2018.
[39] H.-C. Zhang and X.-P. Zhu. Ricci curvature on Alexandrov spaces and rigidity theorems. Comm. Anal. Geom., 18(3):503–553, 2010.
[40] V. Zolotov. Sets with small angles in self-contracted curves. ArXiv e-prints, March
2018.

29

(Nina Lebedeva) Steklov Institute of Mathematics, Russian Academy of
Sciences, 27 Fontanka, 191023 St.Petersburg, Russia
E-mail address, Nina Lebedeva: lebed@pdmi.ras.ru(Shin-ichi Ohta) Department of Mathematics, Osaka University, Osaka 5600043, Japan & RIKEN Center for Advanced Intelligence Project (AIP),
1-4-1 Nihonbashi, Tokyo 103-0027, Japan
E-mail address, Shin-ichi Ohta: s.ohta@math.sci.osaka-u.ac.jp(Vladimir Zolotov) Steklov Institute of Mathematics, Russian Academy of
Sciences, 27 Fontanka, 191023 St.Petersburg, Russia and Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O., 29B, Saint Petersburg 199178 Russia
E-mail address, Vladimir Zolotov: paranuel@mail.ru

30