Regularity of the distance function from arbitrary closed sets

Abstract

We investigate the distance function $$\varvec{\delta }_{K}^{\phi }$$ δ K ϕ from an arbitrary closed subset K of a finite-dimensional Banach space $$ (\mathbf {R}^{n}, \phi ) $$ ( R n , ϕ ) , equipped with a uniformly convex $$ \mathscr {C}^{2} $$ C 2 -norm $$ \phi $$ ϕ . 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, 10.1016/S0723-0869(01)80025-6). We prove that the gradient of $$\varvec{\delta }_{K}^{\phi }$$ δ K ϕ satisfies a Lipschitz property on the complement of the $$\phi $$ ϕ -cut-locus of K (a.k.a. the medial axis of $$\mathbf {R}^{n} {{\,\mathrm{\sim }\,}}K$$ R n ∼ K ) and we prove a structural result for the set of points outside K where $$\varvec{\delta }_{K}^{\phi }$$ δ K ϕ is pointwise twice differentiable, providing an answer to a question raised by Hiriart-Urruty (Am Math Mon 89:456–458, 1982, 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.

Delta‐convex structure of the singular set of distance functions

“…General theory is developed [11,22,40,44,46,57] and it is revealed that the wildness of the ridge is quite sensitively affected by the irregularity of 𝑁. Other types of singular sets and their relations are also under investigation by many authors [12,20,28,39,56].…”

A Remark on Oka’s Lemma and a Geometric Property of Pseudoconvex Domains

“…The distance function and its signed version are classical objects of study. Unfortunately, the basic facts of the associated theory are scattered in the literature -we refer to [2,5,6,13,15,17,19,20,22] for a (necessarily incomplete) list of old and new contributions. As the distance function does not depend on the complex structure, we identify C n with R m for m = 2n in this section and recall the relevant definitions and facts in the real setting: Definition 4 Let ⊂ R m , = R m be a domain.…”