Regularity of distance functions from arbitrary closed sets

TL;DR

This paper studies the regularity properties of distance functions from arbitrary closed sets in Minkowski spaces with uniformly convex norms, establishing Lipschitz continuity of gradients and structural differentiability results.

Contribution

It proves Lipschitz regularity of the gradient of distance functions and characterizes points of twice differentiability, extending classical results to low-regularity and Minkowski space settings.

Findings

Gradient of distance function is Lipschitz outside the cut-locus.

Characterization of points where the distance function is twice differentiable.

Provides sharp generalizations of classical distance function results.

Abstract

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.