Regularity of the distance function to the boundary

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

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