On Sets in Rd with DC Distance Function

We study closed sets F⊂Rd whose distance function dF≔dist(⋅,F) is DC (i.e., is the difference of two convex functions on Rd). Our main result asserts that if F⊂R2 is a graph of a DC function g:R→R, then F has the above property. If d>1, the same holds if g:Rd−1→R is semiconcave, however the case of a general DC function g remains open.