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Large Poisson-Voronoi cells and Crofton cells

Part of:General convexityGeometric probability and stochastic geometry

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 geometry52A40: Inequalities and extremum problems

Information

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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References

[1] Calka, P. (2002). The distributions 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
[2] Gardner, R. J. and Vassallo, S. (1999). Stability of inequalities in the dual Brunn–Minkowski theory. J. Math. Anal. Appl. 231, 568–587.Google Scholar
[3] Goldman, A. (1998). Sur une conjecture de D. G. Kendall concernant la cellule de Crofton du plan et sur sa contrepartie brownienne. Ann. Prob. 26, 1727–1750.CrossRefGoogle Scholar
[4] Groemer, H. (1996). Geometric Applications of Fourier Series and Spherical Harmonics (Encyclopedia Math. Appl. 61). Cambridge University Press.Google Scholar
[5] Groemer, H. and Schneider, R. (1991). Stability estimates for some geometric inequalities. Bull. London Math. Soc. 23, 67–74.CrossRefGoogle Scholar
[6] Hug, D. and Schneider, R. (2004). Large cells in Poisson–Delaunay tessellations. Discrete Comput. Geom. 31, 503–514.Google Scholar
[7] Hug, D., Reitzner, M. and Schneider, R. (2004). The limit shape of the zero cell in a stationary Poisson hyperplane tessellation. Ann. Prob. 32, 1140–1167.CrossRefGoogle Scholar
[8] Kovalenko, I. N. (1997). A proof of a conjecture of David Kendall on the shape of random polygons of large area. Kibernet. Sistem. Anal. 1997, 3–10, 187 (in Russian). English translation: Cybernet. Systems Anal. 33, 461–467.Google Scholar
[9] Kovalenko, I. N. (1998). An extension of a conjecture of D. G. Kendall concerning shapes of random polygons to Poisson Voronoï cells. Proc. Inst. Math. Nat. Acad. Sci. Ukr. Math. Appl. 212, 266–274 (in Ukranian). English translation: in Voronoï's Impact on Modern Science , eds Engel, P. et al., Institute of Mathematics, Kyiv, pp. 266–274.Google Scholar
[10] Kovalenko, I. N. (1999). A simplified proof of a conjecture of D. G. Kendall concerning shapes of random polygons. J. Appl. Math. Stoch. Anal. 12, 301–310.CrossRefGoogle Scholar
[11] Miles, R. E. (1995). A heuristic proof of a long-standing conjecture of D. G. Kendall concerning the shapes of certain large random polygons. Adv. Appl. Prob. 27, 397–417.Google Scholar
[12] Möller, J., (1994). Lectures on Random Voronoi Tessellations (Lecture Notes Statist. 87). Springer, New York.Google Scholar
[13] Möller, J., (1998). A review on probabilistic models and results for Voronoi tessellations. Proc. Inst. Math. Nat. Acad. Sci. Ukr. Math. Appl. 212, 254–265 (in Ukranian). English translation: in Voronoï's Impact on Modern Science , eds Engel, P. et al., Institute of Mathematics, Kyiv, pp. 254–265.Google Scholar
[14] Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations; Concepts and Applications of Voronoi Diagrams, 2nd edn. John Wiley, Chichester.Google Scholar
[15] Schneider, R. (1993). Convex Bodies: the Brunn–Minkowski Theory (Encyclopedia Math. Appl. 44). Cambridge University Press.Google Scholar
[16] Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Stuttgart.Google Scholar
[17] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar