Polynomial parallel volume, convexity and contact distributions of random sets

Abstract

We characterize convexity of a random compact set X in ℝd via polynomial expected parallel volume of X. The parallel volume of a compact set A is a function of r ≥ 0 and is defined here in two steps. First we form the parallel set at distance r with respect to a one- or two-dimensional gauge body B. Then we integrate the volume of this (relative) parallel set with respect to all rotations of B. We apply our results to characterize convexity of the typical grain of a Boolean model via first contact distributions.