Quasi-convex subsets in Alexandrov spaces with lower curvature bound

TL;DR

This paper introduces quasi-convex subsets in Alexandrov spaces with lower curvature bounds, exploring their properties and significance in understanding the geometric structure compared to Riemannian manifolds.

Contribution

It defines quasi-convex subsets in Alexandrov spaces, including extremal and convex subsets, and generalizes the Liberman theorem to these spaces.

Findings

Quasi-convex subsets include all closed convex subsets without boundary and extremal subsets.

The paper establishes fundamental properties of quasi-convex subsets.

A generalized Liberman theorem for Alexandrov spaces is proved.

Abstract

In this paper, we introduce quasi-convex subsets in Alxandrov spaces with lower curvature bound, which include not only all closed convex subsets without boundary but also all extremal subsets. Moreover, we explore several essential properties of such kind of subsets including a generalized Liberman theorem. It turns out that the quasi-convex subset is a nice and fundamental concept to illustrate the similarities and differences between Riemannian manifolds and Alxandrov spaces with lower curvature bound.