Regularity of the distance function to the boundary

Abstract

Let $\Omega$ be a domain in a smooth complete Finsler manifold, and let $G$ be the largest open subset of $\Omega$ such that for every $x$ in $G$ there is a unique closest point from $\partial \Omega$ to $x$ (measured in the Finsler metric). We prove that the distance function from $\partial \Omega$ is in $C^{k,\alpha}_{loc}(G\cup \partial \Omega)$, $k\ge 2$ and $0<\alpha\le 1$, if $\partial \Omega$ is in $C^{k,\alpha}$.