A characterization of cut locus for C1 hypersurfaces

Abstract

Let Ω be an open set in Rn with C1-boundary and Σ be the skeleton of Ω , which consists of points where the distance function to ∂Ω is not differentiable. This paper characterizes the cut locus (ridge) Σ ¯ , 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 Σ ¯.