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Large Poisson-Voronoi cells and Crofton cells

Part of: Geometric probability and stochastic geometry General convexity

Published online by Cambridge University Press:  01 July 2016

Daniel Hug ,
Matthias Reitzner  and
Rolf Schneider
Daniel Hug*
Affiliation:
Albert-Ludwigs-Universität Freiburg
Matthias Reitzner*
Affiliation:
Technische Universität Wien
Rolf Schneider*
Affiliation:
Albert-Ludwigs-Universität Freiburg
*
∗ Postal address: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstr. 1, D-79104 Freiburg, Germany.
∗∗∗ Postal address: Institut für Analysis und Technische Mathematik, Technische Universität Wien, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria. Email address: mreitzne@mail.zserv.tuwien.ac.at
∗∗∗∗ Email address: rolf.schneider@math.uni-freiburg.de
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Abstract

It is proved that the shape of the typical cell of a stationary Poisson-Voronoi tessellation in Euclidean space, under the condition that the volume of the typical cell is large, must be close to spherical, with high probability. The same holds if the volume is replaced by the surface area or other suitable functionals. Similar results are established for the zero cell of a stationary and isotropic Poisson hyperplane tessellation.

Keywords

Poisson-Voronoi tessellationtypical cellPoisson hyperplane tessellationCrofton cellasymptotic shapeD. G. Kendall's conjectureintrinsic volume

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry 52A40: Inequalities and extremum problems
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 36 , Issue 3 , September 2004 , pp. 667 - 690
Copyright
Copyright © Applied Probability Trust 2004 

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