The Annals of Probability

The limit shape of the zero cell in a stationary Poisson hyperplane tessellation

Daniel Hug, Matthias Reitzner, and Rolf Schneider

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Abstract

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.

Article information

Source
Ann. Probab., Volume 32, Number 1B (2004), 1140-1167.

Dates
First available in Project Euclid: 11 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1079021474

Digital Object Identifier
doi:10.1214/aop/1079021474

Mathematical Reviews number (MathSciNet)
MR2044676

Zentralblatt MATH identifier
1050.60010

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05]

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

Citation

Hug, Daniel; Reitzner, Matthias; Schneider, Rolf. The limit shape of the zero cell in a stationary Poisson hyperplane tessellation. Ann. Probab. 32 (2004), no. 1B, 1140--1167. doi:10.1214/aop/1079021474. https://projecteuclid.org/euclid.aop/1079021474


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References

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