Generalized contact distributions of inhomogeneous Boolean models

Hug, Daniel
Last, Günter
Weil, Wolfgang

facul_01; 09

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 $L\subset \R^d$ and a gauge body $B\subset \R^d$ 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 $Z\cap L=\emptyset$,
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 $l_B(L,\cdot)$ may be taken from a large class of locally defined
functionals. In particular, we pursue the question to which extent the spatial
density and the grain distribution underlying an inhomogeneous Boolean model $Z$
are determined by the generalized contact distributions of $Z$.

Primary MSC: 60D05 Geometric probability, stochastic geometry, random sets, 60G57 Random measures, 52A21 Finite-dimensional Banach spaces (including special norms, zonoids, etc.)
Secondary MSC: 60G55 Point processes, 52A22 Random convex sets and integral geometry, 52A20 Convex sets in $n$ dimensions (including convex hypersurfaces), 53C65 Integral geometry, 46B20 Geometry and structure of normed linear spaces
Keywords: Stochastic geometry, contact distribution function, germ-grain model, Boolean model, curvature measure, marked point process, Palm probabilities, random measure