# On sets in ${\mathbb R}^d$ with DC distance function

## TL;DR

This paper investigates the geometric properties of closed sets in Euclidean space whose distance functions are DC, establishing that in two dimensions, graphs of DC functions have DC distance functions, with partial results in higher dimensions.

## Contribution

The paper proves that in two dimensions, graphs of DC functions have DC distance functions, and extends this to higher dimensions for semiconcave functions, highlighting open problems for general DC functions.

## Key findings

- Graphs of DC functions in ${f R}^2$ have DC distance functions.
- In higher dimensions, semiconcave functions also produce sets with DC distance functions.
- The case for general DC functions in higher dimensions remains unresolved.

## Abstract

We study closed sets $F \subset {\mathbb R}^d$ whose distance function $d_F:= {\rm dist}\,(\cdot,F)$ is DC (i.e., is the difference of two convex functions on ${\mathbb R}^d$). Our main result asserts that if $F \subset {\mathbb R}^2$ is a graph of a DC function $g:{\mathbb R}\to {\mathbb R}$, then $F$ has the above property. If $d>1$, the same holds if $g:{\mathbb R}^{d-1}\to {\mathbb R}$ is semiconcave, however the case of a general DC function $g$ remains open.