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Poisson hyperplane tessellation: Asymptotic probabilities of the zero and typical cells

We consider the distribution of the zero and typical cells of a (homogeneous) Poisson hyperplane tessellation. We give a direct proof adapted to our setting of the well known Complementary Theorem. We provide sharp bounds for the tail distribution of the number of facets. We also improve existing bounds for the tail distribution of size measurements of the cells, such as the volume or the mean width. We improve known results about the generalised D.G. Kendall's problem, which asks about the shape of large cells. We also show that cells with many facets cannot be close to a lower dimensional convex body. We tacle the much less study problem of the number of facets and the shape of small cells. In order to obtain the results above we also develop some purely geometric tools, in particular we give new results concerning the polytopal approximation of an elongated convex body.

Identiferoai:union.ndltd.org:uni-osnabrueck.de/oai:repositorium.ub.uni-osnabrueck.de:urn:nbn:de:gbv:700-2017021715545
Date17 February 2017
CreatorsBonnet, Gilles
ContributorsProf. Dr. Matthias Reitzner, Prof. Dr. Ilya Molchanov
Source SetsUniversität Osnabrück
LanguageEnglish
Detected LanguageEnglish
Typedoc-type:doctoralThesis
Formatapplication/pdf, application/zip
RightsNamensnennung - Weitergabe unter gleichen Bedingungen 3.0 Unported, http://creativecommons.org/licenses/by-sa/3.0/

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