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Random Laguerre tessellations

Part of: Designs and configurations Stochastic processes General convexity Multivariate analysis Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Claudia Lautensack  and
Sergei Zuyev
Claudia Lautensack*
Affiliation:
University of Applied Sciences, Darmstadt
Sergei Zuyev*
Affiliation:
University of Strathclyde
*
∗ Postal address: Fraunhofer ITWM, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany. Email address: claudia.lautensack@itwm.fraunhofer.de
∗∗ Postal address: Department of Statistics and Modelling Science, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK. Email address: sergei@stams.strath.ac.uk
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Abstract

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A systematic study of random Laguerre tessellations, weighted generalisations of the well-known Voronoi tessellations, is presented. We prove that every normal tessellation with convex cells in dimension three and higher is a Laguerre tessellation. Tessellations generated by stationary marked Poisson processes are then studied in detail. For these tessellations, we obtain integral formulae for geometric characteristics and densities of the typical k-faces. We present a formula for the linear contact distribution function and prove various limit results for convergence of Laguerre to Poisson-Voronoi tessellations. The obtained integral formulae are subsequently evaluated numerically for the planar case, demonstrating their applicability for practical purposes.

Keywords

Laguerre tessellationpower tessellationweighted Voronoi tessellationk-face densityPoisson processrandom tessellationstochastic geometry

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Secondary: 05B45: Tessellation and tiling problems 52A22: Random convex sets and integral geometry 62H11: Directional data; spatial statistics
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 40 , Issue 3 , September 2008 , pp. 630 - 650
Copyright
Copyright © Applied Probability Trust 2008 

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