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 closed locally WDC set in {\mathbb R}^2 is a WDC set. Another consequence is that compact WDC subsets of {\mathbb R}^2 form a Borel subset of the space of all compact sets.
Keywords: distance function; WDC set; DC function; DC aura; Borel complexity
DOI: DOI 10.14712/1213-7243.2021.006
AMS Subject Classification: 26B25