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Densities of mixed volumes for Boolean models

Part of: General convexity Stochastic processes Geometric probability and stochastic geometry Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Wolfgang Weil
Wolfgang Weil*
Affiliation:
Universität Karlsruhe
*
∗ Postal address: Mathematisches Institut II, Universität Karlsruhe, D-76128 Karlsruhe, Germany. Email address: [email protected]
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Abstract

In generalization of the well-known formulae for quermass densities of stationary and isotropic Boolean models, we prove corresponding results for densities of mixed volumes in the stationary situation and show how they can be used to determine the intensity of non-isotropic Boolean models Z in d-dimensional space for d = 2, 3, 4. We then consider non-stationary Boolean models and extend results of Fallert on quermass densities to densities of mixed volumes. In particular, we present explicit formulae for a planar inhomogeneous Boolean model with circular grains.

Keywords

Boolean modelstationarymixed volumesPoisson processintensityinhomogeneous

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A39: Mixed volumes and related topics 60G55: Point processes 62M30: Spatial processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 33 , Issue 1 , March 2001 , pp. 39 - 60
Copyright
Copyright © Applied Probability Trust 2001 

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