On support measures in Minkowski spaces and contact distributions in stochastic geometry

Abstract

This paper is concerned with contact distribution functions of a random closed set $\Xi=\Bigcup_{n=1}^\infty \Xi_n$ in $\mathbb{R}^d$, where the $\Xi_n$ are assumed to be random nonempty convex bodies. These distribution functions are defined here in terms of a distance function which is associated with a strictly convex gauge body (structuring element) that contains the origin in its interior. Support measures with respect to such distances will be introduced and extended to sets in the local convex ring.These measures will then be used in a systematic way to derive and describe some of the basic properties of contact distribution functions. Most of the results are obtained in a general nonstationary setting.Only the final section deals with the stationary case.