# Self-contracted curves in spaces with weak lower curvature bound

## TL;DR

This paper proves that bounded self-contracted curves are rectifiable in a broad class of metric spaces with weak lower curvature bounds, extending previous results to include spaces with lower curvature bounds and analyzing snowflake embeddings.

## Contribution

It introduces a new class of metric spaces with weak lower curvature bounds and demonstrates rectifiability of self-contracted curves within this class, extending prior work on upper curvature bounds.

## Key findings

- Bounded self-contracted curves are rectifiable in these spaces.
- Large snowflakes cannot be embedded into these metric spaces.
- The strategy used for spaces with upper curvature bounds applies to lower curvature bound spaces.

## 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, our condition 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) metric spaces 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 bound as well.