Regularity of the distance function to the boundary

TL;DR

This paper proves that in a smooth Finsler manifold, the distance function to the boundary is locally smooth up to the boundary, provided the boundary itself is sufficiently smooth.

Contribution

It establishes the regularity of the distance function in Finsler geometry, extending classical results to more general metric spaces.

Findings

Distance function is in $C^{k,eta}_{loc}$ on the set where closest boundary points are unique.

Regularity of the distance function matches the boundary's smoothness level.

Results apply to smooth complete Finsler manifolds with smooth boundaries.

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 $C^{k,\alpha}$.