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January 2004 The limit shape of the zero cell in a stationary Poisson hyperplane tessellation
Daniel Hug, Matthias Reitzner, Rolf Schneider
Ann. Probab. 32(1B): 1140-1167 (January 2004). DOI: 10.1214/aop/1079021474


In the early 1940s, D. G. Kendall conjectured that the shape of the zero cell of the random tessellation generated by a stationary and isotropic Poisson line process in the plane tends to circularity given that the area of the zero cell tends to $\infty$. A proof was given by I. N. Kovalenko in 1997. This paper generalizes Kovalenko's result in two directions: to higher dimensions and to not necessarily isotropic stationary Poisson hyperplane processes. In the anisotropic case, the asymptotic shape of the zero cell depends on the direction distribution of the hyperplane process and is obtained from it via an application of Minkowski's existence theorem for convex bodies with given area measures.


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Daniel Hug. Matthias Reitzner. Rolf Schneider. "The limit shape of the zero cell in a stationary Poisson hyperplane tessellation." Ann. Probab. 32 (1B) 1140 - 1167, January 2004.


Published: January 2004
First available in Project Euclid: 11 March 2004

zbMATH: 1050.60010
MathSciNet: MR2044676
Digital Object Identifier: 10.1214/aop/1079021474

Primary: 60D05
Secondary: 52A22

Keywords: asymptotic shape , Crofton cell , D. G. Kendall's conjecture , hyperplane tessellation , Poisson hyperplane process , typical cell , zero cell

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 1B • January 2004
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