Regularity of the distance function from arbitrary closed sets

Abstract

We investigate the distance function varvec{delta }_{K}^{phi } from an arbitrary closed subset K of a finite-dimensional Banach space (mathbf {R}^{n}, phi ) , equipped with a uniformly convex mathscr {C}^{2} -norm phi . These spaces are known as Minkowski spaces and they are one of the fundamental spaces of Finslerian geometry (see Martini et al. in Expo Math 19:97–142, 2001, https://doi.org/10.1016/S0723-0869(01)80025-6). We prove that the gradient of varvec{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} {{,mathrm{sim },}}K) and we prove a structural result for the set of points outside K where varvec{delta }_{K}^{phi } is pointwise twice differentiable, providing an answer to a question raised by Hiriart-Urruty (Am Math Mon 89:456–458, 1982, 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.