  We study WDC sets, which form a substantial generalization of sets with
positive reach and still admit the definition of curvature measures. Main
results concern WDC sets $A\subset \mathbb{R}^2$. We prove that, for such $A$,
the distance function $d_A= {\rm dist}(\cdot,A)$ is a `DC aura' for $A$, which
implies that each locally WDC set in $\mathbb{R}^2$ is a WDC set. An another
consequence is that compact WDC subsets of $\mathbb{R}^2$ form a Borel subset
of the space of all compact sets.
