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The distributions of the smallest disks containing the Poisson-Voronoi typical cell and the Crofton cell in the plane

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Pierre Calka
Pierre Calka*
Affiliation:
Université Claude Bernard Lyon 1
*
∗ Postal address: Université Claude Bernard Lyon 1, LaPCS, Bâtiment B, Domaine de Gerland, 50 avenue Tony-Garnier, F-69366 Lyon Cedex 07, France. Email address: pierre.calka@univ-lyon1.fr
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Abstract

Among the disks centered at a typical particle of the two-dimensional Poisson-Voronoi tessellation, let Rm be the radius of the largest included within the polygonal cell associated with that particle and RM be the radius of the smallest containing that polygonal cell. In this article, we obtain the joint distribution of Rm and RM. This result is derived from the covering properties of the circle due to Stevens, Siegel and Holst. The same method works for studying the Crofton cell associated with the Poisson line process in the plane. The computation of the conditional probabilities P{RM ≥ r + s | Rm = r} reveals the circular property of the Poisson-Voronoi typical cells (as well as the Crofton cells) having a ‘large’ in-disk.

Keywords

Coverage of the circleCrofton cellPalm distributionPoisson line processPoisson-Voronoi tessellationstochastic geometrytypical cell

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 34 , Issue 4 , December 2002 , pp. 702 - 717
Copyright
Copyright © Applied Probability Trust 2002 

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