Large faces in Poisson hyperplane mosaics

Daniel Hug; Rolf Schneider

doi:10.1214/09-AOP510

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Home > Journals > Ann. Probab. > Volume 38 > Issue 3 > Article

May 2010 Large faces in Poisson hyperplane mosaics

Daniel Hug, Rolf Schneider

Ann. Probab. 38(3): 1320-1344 (May 2010). DOI: 10.1214/09-AOP510

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Abstract

A generalized version of a well-known problem of D. G. Kendall states that the zero cell of a stationary Poisson hyperplane tessellation in ℝ^d, 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.