Self-Contracted Curves in Spaces With Weak Lower Curvature Bound

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. Here we apply this strategy to spaces with lower curvature bound.

Automated summary

In this article, we show that bounded self-contracted curves are rectifiable in metric spaces with weak lower curvature bound that includes finite-dimensional Alexandrov spaces of curvature bounded below and Berwald spaces of nonnegative flag curvature. Our condition, regarded as a strengthened doubling condition, also holds for certain metric spaces with upper curvature bound. Additionally, we demonstrate the non-embeddability of large snowflakes into metric spaces in the same class, following a strategy previously applied to spaces with upper curvature bound.