Regularity of distance functions from arbitrary closed sets

We investigate the distance function $\boldsymbol{\delta}_{K}^{\phi}$ from an arbitrary closed subset $K$ of a~finite-dimensional Banach space $(\mathbf{R}^{n}, \phi)$, equipped with a uniformly convex $\mathcal{C}^{2}$-norm $\phi$. These spaces are known as \emph{Minkowski spaces} and they are one of the fundamental spaces of Finslerian geometry (see https://doi.org/10.1016/S0723-0869(01)80025-6). We prove that the gradient of $\boldsymbol{\delta}_{K}^{\phi}$ satisfies a Lipschitz property on the complement of the $\phi$-cut-locus of $K$ (a.k.a. the medial axis of $\mathbf{R}^{n} \sim K$) and we prove a~structural result for the set of points outside $K$ where $\boldsymbol{\delta}_{K}^{\phi}$ is pointwise twice differentiable, providing an answer to a question raised by Hiriart-Urruty (see https://doi.org/10.2307/2321379). Our results give sharp generalisations of some classical results in the theory of distance functions and they are motivated by critical low-regularity examples for which the available results gives no meaningful or very restricted informations. The results of this paper find natural applications in the theory of partial differential equations and in convex geometry.