A characterization of sets in $\mathbb{R}^2$ with DC distance function

TL;DR

This paper provides a complete characterization of closed sets in the plane whose distance functions are DC functions, revealing new properties of these sets and advancing understanding of their geometric structure.

Contribution

It introduces a full characterization of closed sets in \,\mathbb{R}^2\ that have DC distance functions, a novel geometric insight.

Findings

Characterization of sets with DC distance functions

Properties of sets with DC distance functions established

Enhanced understanding of geometric structure of such sets

Abstract

We give a complete characterization of closed sets $F \subset \mathbb{R}^2$ whose distance function $d_F:= \mathrm{dist}(\cdot,F)$ is DC (i.e., is the difference of two convex functions on $\mathbb{R}^2$). Using this characterization, a number of properties of such sets is proved.