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

Abstract

We study closed sets $F\subset {\mathbb{R}}^d$ whose distance function $d_F:=\mathrm{dist}(\cdot ,F)$ 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.

Some minor errors were corrected and the last section was considerably extended