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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 functionson ${\mathbb R}^d$). 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 theabove property. If $d>1$, the same holds if $g:{\mathbb R}^{d-1}\to {\mathbbR}$ is semiconcave, however the case of a general DC function $g$ remains open.

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