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

Tatsuya Miura

Published 2015-09-02Version 1

Let $\Omega$ be an open set in $\mathbb{R}^n$ with $C^1$-boundary. Let $\Sigma$ be the skeleton of $\Omega$, that is, the set $\Sigma$ 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 $\Sigma$, by introducing the lower semicontinuous envelope of radius of curvature of the boundary $\partial\Omega$.