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The sectional Poisson Voronoi tessellation is not a Voronoi tessellation

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

Published online by Cambridge University Press:  01 July 2016

S. N. Chiu ,
R. Van De Weygaert  and
D. Stoyan
S. N. Chiu
Affiliation:
TU Bergakademie Freiburg
R. Van De Weygaert
Affiliation:
Canadian Institute for Astrophysics, Sterrewacht Leiden and Max Planck Institute für Astrophysik
D. Stoyan*
Affiliation:
TU Bergakademie Freiburg
*
∗∗∗ Postal address: TU Bergakademie Freiburg, Institut für Stochastik, 09596 Freiburg, Germany.
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Abstract

Is the intersection between an arbitrary but fixed plane and the spatial Poisson Voronoi tessellation a planar Voronoi tessellation? In this paper a negative answer is given to this long-standing question in stochastic geometry. The answer remains negative for the intersection between a t-dimensional linear affine space and the d-dimensional Poisson Voronoi tesssellation, where 2 ≦ t ≦ d − 1. Moreover, it is shown that each cell on this intersection is almost surely a non-Voronoi cell.

Keywords

POISSON PROCESSSECTIONAL VORONOI TESSELLATIONSTOCHASTIC GEOMETRYVORONOI TESSELLATION

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Secondary: 52A22: Random convex sets and integral geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 28 , Issue 2 , June 1996 , pp. 356 - 376
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

∗

Present address: Department of Mathematics, Hong Kong Baptist University, Waterloo Road, Kowloon, Hong Kong.

∗∗

Present address: Kapteyn Instituut, University of Groningen, Postbus 800, 9700 AV Groningen, The Netherlands.

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