Convexity, critical points, and connectivity radius

TL;DR

This paper investigates the geometric properties of level sets of the distance function from a boundary point in convex sets, establishing bounds on their connectivity based on critical points.

Contribution

It provides a lower bound for the connectivity radius of level sets in convex sets using the critical points of the distance function.

Findings

Lower bound for connectivity radius in convex sets

Relationship between critical points and level set connectivity

Insights into the structure of convex set boundary level sets

Abstract

We study the level sets of the distance function from a boundary point of a convex set in Euclidean space. We provide a lower bound for the range of connectivity of the level sets, in terms of the critical points of the distance function in the sense of Grove-Shiohama-Gromov-Cheeger.