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1	Random Polytopes Beermann, Mareen 23 June 2015 (has links) Random polytopes can be constructed in many different ways. In this thesis two certain kinds are considered - random polytopes as the convex hull of random points and as the intersection of finitely many random half spaces. Concerning these two models different issues are treated. Random Polytope Random Tessellation 31.70 - Wahrscheinlichkeitsrechnung 31.50 - Geometrie: Allgemeines ddc:510
2	Poisson hyperplane tessellation: Asymptotic probabilities of the zero and typical cells Bonnet, Gilles 17 February 2017 (has links) 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. Stochastic Geometry Convex Geometry tessellation mosaic random tessellation Poisson hyperplane tessellation Zero cell Typical cell Asymptotic probabilities D.G. Kendall's problem Complementary theorem polytopal approximation delta-net elongated convex bodies facets Phi-Content center shape tail distribution small cells ddc:510

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Random Polytopes

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