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

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 $\overline{\Sigma}$.