The Annals of Probability

Large faces in Poisson hyperplane mosaics

Daniel Hug and Rolf Schneider

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A generalized version of a well-known problem of D. G. Kendall states that the zero cell of a stationary Poisson hyperplane tessellation in ℝd, under the condition that it has large volume, approximates with high probability a certain definite shape, which is determined by the directional distribution of the underlying hyperplane process. This result is extended here to typical k-faces of the tessellation, for k∈{2, …, d−1}. This requires the additional condition that the direction of the face be in a sufficiently small neighbourhood of a given direction.

Article information

Ann. Probab., Volume 38, Number 3 (2010), 1320-1344.

First available in Project Euclid: 2 June 2010

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45]

Poisson hyperplane tessellation volume-weighted typical face D. G. Kendall’s problem limit shape


Hug, Daniel; Schneider, Rolf. Large faces in Poisson hyperplane mosaics. Ann. Probab. 38 (2010), no. 3, 1320--1344. doi:10.1214/09-AOP510.

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