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

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 $\overline{\Sigma }$.

12 pages, 3 figures

References in corpus (3)

• On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: Regularity and geometric results
• Characterization of stadium-like domains via boundary value problems for the infinity Laplacian
• A rigorous setting for the reinitialization of first order level set equations

Cited by in corpus (3)

• Distance functions with dense singular sets
• The tangent cone, the dimension and the frontier of a medial axis
• Delta-convex structure of the singular set of distance functions