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On the distribution of the spherical contact vector of stationary germ-grain models

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

G. Last  and
R. Schassberger
G. Last*
Affiliation:
Technical University of Braunschweig
R. Schassberger*
Affiliation:
Technical University of Braunschweig
*
∗ Postal address: Institut für Mathematische Stochastik, Technische Universität Braunschweig, Pockelsstraße 14, Postfach 33,29, 38023 Braunschweig, Germany.
∗ Postal address: Institut für Mathematische Stochastik, Technische Universität Braunschweig, Pockelsstraße 14, Postfach 33,29, 38023 Braunschweig, Germany.
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Abstract

We consider a stationary germ-grain model Ξ with convex and compact grains and the distance r(x) from x ε ℝd to Ξ. For almost all points x ε ℝd there exists a unique point p(x) in the boundary of Ξ such that r(x) is the length of the vector x-p(x), which is called the spherical contact vector at x. In this paper we relate the distribution of the spherical contact vector to the times it takes a typical boundary point of Ξ to hit another grain if all grains start growing at the same time and at the same speed. The notion of a typical point is made precise by using the generalized curvature measures of Ξ. The result generalizes a well known formula for the Boolean model. Specific examples are discussed in detail.

Keywords

Spherical contact vectorspherical contact distributioncurvature measuresgerm-grain modelrandom settessellationrandom measurePalm probability

MSC classification

Primary: 60G57: Random measures
Secondary: 60D05: Geometric probability and stochastic geometry
Type
Stochatic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 30 , Issue 1 , March 1998 , pp. 36 - 52
Copyright
Copyright © Applied Probability Trust 1998 

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