  WDC sets in ${\mathbb R}^d$ were recently defined as sublevel sets of DC
functions (differences of convex functions) at weakly regular values. They form
a natural and substantial generalization of sets with positive reach and still
admit the definition of curvature measures. Using results on singularities of
convex functions, we obtain regularity results on the boundaries of WDC sets.
In particular, the boundary of a compact WDC set can be covered by finitely
many DC surfaces. More generally, we prove that any compact WDC set $M$ of
topological dimension $k\leq d$ can be decomposed into the union of two sets,
one of them being a $k$-dimensional DC manifold open in $M$, and the other can
be covered by finitely many DC surfaces of dimension $k-1$. We also
characterize locally WDC sets among closed Lipschitz domains and among
lower-dimensional Lipschitz manifolds. Finally, we find a full characterization
of locally WDC sets in the plane.
