The Annals of Probability

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

Daniel Hug and Günter Last

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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.

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Ann. Probab., Volume 28, Number 2 (2000), 796-850.

First available in Project Euclid: 18 April 2002

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Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G57: Random measures 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx]
Secondary: 60G55: Point processes 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx] 46B20: Geometry and structure of normed linear spaces

Stochastic geometry Minkowski space contact distribution function germ-grain model support (curvature) measure marked point process Palm probabilities randommeasure


Hug, Daniel; Last, Günter. On support measures in Minkowski spaces and contact distributions in stochastic geometry. Ann. Probab. 28 (2000), no. 2, 796--850. doi:10.1214/aop/1019160261.

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