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On the correlation between the volumes of the typical Poisson-Voronoi cell and the typical Stienen sphere

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Viktor Olsbo
Viktor Olsbo*
Affiliation:
Chalmers University of Technology and Göteborg University
*
∗ Postal address: Mathematical Sciences, Chalmers University of Technology and Göteborg University, SE-412 96 Göteborg, Sweden. Email address: vikol@chalmers.se
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Abstract

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In this paper we consider a tessellation V generated by a homogeneous Poisson process Φ in Rd and, furthermore, the random set of spheres with centres being the points in Φ and having radii equal to half the distance to their closest neighbouring point in Φ. In Rd we give an integral formula for the correlation between the volume of the typical cell and the volume of the sphere in the typical cell, and we also show that this correlation is strictly positive. Furthermore, on the real line we give an analytical expression for the correlation, and in the plane and in space we give simplified integral formulae. Numerical values for the correlation for d = 2,…,7 are also given.

Keywords

CorrelationPoisson processRobbins' formulaStienen modeltypical cellVoronoi tessellation

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 39 , Issue 4 , December 2007 , pp. 883 - 892
Copyright
Copyright © Applied Probability Trust 2007 

References

Calka, P. (2002). The distribution of the smallest disks containing the Poisson–Voronoi typical cell and the Crofton cell in the plane. Adv. Appl. Prob. 34, 702–717.CrossRefGoogle Scholar
Calka, P. (2003). Precise formulae for the distributions of the principal geometric characteristics of the typical cells of a two-dimensional Poisson–Voronoi tessellation and a Poisson line process. Adv. Appl. Prob. 35, 551–562.CrossRefGoogle Scholar
Gilbert, E. N. (1962). Random subdivisions of space into crystals. Ann. Math. Statist. 33, 958–972.Google Scholar
Hug, D., Reitzner, M. and Schneider, R. (2002). Large Poisson–Voronoi cells and Crofton cells. Adv. Appl. Prob. 34, 702–717.Google Scholar
Meijering, J. L. (1953). Interface area, edge length and number of vertices in crystal aggregates with random nucleation. Philips Res. Rep. 8, 270–290.Google Scholar
Møller, J. (1994). Lectures on Random Voronoi Tessellations. Springer, New York.CrossRefGoogle Scholar
Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations, Concepts and Applications of Voronoi Diagrams, 2nd edn. John Wiley, Chichester.CrossRefGoogle Scholar
Robbins, H. E. (1944). On the measure of a random set. I. Ann. Math. Statist. 15, 70–74.Google Scholar
Robbins, H. E. (1945). On the measure of a random set. II. Ann. Math. Statist. 16, 342–347.CrossRefGoogle Scholar
Schlater, M. and Stoyan, D. (1997). The covariance of the Stienen model. In Advances in Theory and Applications of Random Sets, ed. Jeulin, D., World Scientific, Singapore, pp. 157–174.Google Scholar
Stienen, H. (1982). Die Vergröberung von Karbiden in reinen Eisen–Kohlenstoff Stählen. , RWTH Aachen.Google Scholar