# SRA-free condition by Zolotov for self-contracted curves and nondegeneracy of zz-distance for M\"obius structures on the circle

## TL;DR

This paper introduces a Moebius invariant SRA-free condition to analyze self-contracted curves and demonstrates that the zz-distance in Moebius structures on the circle is nondegenerate, aiding the inverse problem in Moebius geometry.

## Contribution

It develops a Moebius invariant SRA-free condition and proves the nondegeneracy of zz-distance for Moebius structures on the circle, advancing inverse Moebius geometry.

## Key findings

- zz-distance is nondegenerate for Moebius structures on the circle
- The Moebius invariant SRA-free condition applies to self-contracted curves
- Progress towards solving the inverse problem in Moebius geometry

## Abstract

SRA-free condition for metric spaces (that is, spaces without Small Rough Angles) was introduced by Zolotov to study rectifiability of self-contracted curves in various metric spaces. We give a Moebius invariant version of this notion which allows to show that zz-distance associated with a respective Moebius structure on the circle is nondegenerate. This result is an important part of a solution to the inverse problem of Moebius geometry on the circle.