In stochastic geometry, the model of particle process with convex particles is usually applied when working with characteristics derived from curvature measures. A more general framework is that of particles with positive reach (Federer, 1959); the stochastic model was introduced by Zähle (1986). Recently, a substantially larger family of WDC sets still admitting curvature measures was introduced (Pokorný & R., 2013).
In order to approach stochastic applications, it has to be shown that the family of WDC sets with suitable metric is a standard Borel space.
For the subclass of compact DC domains (sets whose boundary is locally representable as difference of convex functions), this has been shown by Zubal (2015).