Sets with small angles in self-contracted curves

Abstract

We study metric spaces with bounded rough angles. E. Le Donne, T. Rajala and E. Walsberg implicitly used this notion to show that infinite snowflakes can not be isometrically embedded into finite dimensional Banach spaces.We show that bounded non-rectifiable self-contracted curves contain metric subspaces with bounded rough angles. Which provides rectifiability of bounded self-contracted curves in a wide class of metric spaces including reversible C ∞ -Finsler manifolds, locally compact CAT(k)-spaces with locally extendable geodesics and locally compact Busemann spaces with locally extendable geodesics.We also extend the result on non embeddability of infinite snowflakes to this class of spaces.2010 Mathematics Subject Classification. 51F99.

SRA-free condition by Zolotov for self-contracted curves and nondegeneracy of zz-distance for Möbius structures on the circle

“…for some and hence any semi-metric from M . This is a Möbius invariant version of the SRA-free condition introduced by Zolotov in [Zo18]. Now, our main result is as follows.…”

“…Self-contracted curves usually appear as gradient curves of convex functions, and they play an important role in a number of questions. Basic problem for a self-contrated curve is to establish its rectifiability, see [DDDL], [DDDR], [DLS], [LOZ], [Le], [Ohta], [ST], [Zo18].…”

Inverse problem for Moebius geometry on the circle

“…Remark 2.3. Axiom M(α) is motivated by the work [Zo18] of V. Zolotov. It is stronger than that in [Bu19].…”