Generalized contact distributions of inhomogeneous Boolean models

Abstract

The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ R^d and a gauge body B ⊂ R^d, such a generalized contact distribution is the conditional distribution of the random vector (d_B(L,Z),u_B(L,Z),p_B(L,Z),l_B(L,Z)) given that Z∩L = ∅, where Z is a Boolean model, d_B(L,Z) is the distance of L from Z with respect to B, p_B(L,Z) is the boundary point in L realizing this distance (if it exists uniquely), u_B(L,Z) is the corresponding boundary point of B (if it exists uniquely) and l_B(L,·) may be taken from a large class of locally defined functionals. In particular, we pursue the question of the extent to which the spatial density and the grain distribution underlying an inhomogeneous Boolean model Z are determined by the generalized contact distributions of Z.