Remarks on WDC sets

Abstract

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 $d_A=\mathrm{dist}(\cdot ,A)$ 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.