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