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

We give a complete characterization of closed sets $F\subset {\mathbb{R}}^2$ whose distance function $d_F:=\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(\cdot ,F)$ 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.