A characterization of cut locus for $C^1$ hypersurfaces

TL;DR

This paper characterizes the cut locus of $C^1$ hypersurfaces using a generalized radius of curvature and provides conditions for its measure to vanish.

Contribution

It introduces a generalized radius of curvature for $C^1$ hypersurfaces and characterizes the cut locus through this concept.

Findings

The cut locus can be described via a generalized radius of curvature.

A sufficient condition for the measure of the cut locus to be zero is established.

The characterization applies to $C^1$ hypersurfaces with potential implications for geometric analysis.

Abstract

Let $\Omega$ be an open set in $\mathbb{R}^n$ with $C^1$-boundary and $\Sigma$ be the skeleton of $\Omega$, which consists of points where the distance function to $\partial\Omega$ is not differentiable. This paper characterizes the cut locus (ridge) $\overline{\Sigma}$, which is the closure of the skeleton, by introducing a generalized radius of curvature and its lower semicontinuous envelope. As an application we give a sufficient condition for vanishing of the Lebesgue measure of $\overline{\Sigma}$.