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

Abstract: We study closed sets $F \subset {\mathbb R}<sup>d$ whose distance function $d_F:= {\rm dist}\,(\cdot,F)$ is DC (i.e., is the difference of two convex functions on ${\mathbb R}<sup>d$). Our main result asserts that if $F \subset {\mathbb R}<sup>2$ is a graph of a DC function $g:{\mathbb R}\to {\mathbb R}$, then $F$ has the above property. If $d&gt;1$, 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.