Poisson polyhedra in high dimensions |
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Authors: | Julia Hörrmann Daniel Hug Matthias Reitzner Christoph Thäle |
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Institution: | 1. Ruhr University Bochum, Faculty of Mathematics, D-44780 Bochum, Germany;2. Karlsruhe Institute of Technology, Department of Mathematics, Institute of Stochastics, 76128 D-Karlsruhe, Germany;3. Osnabrück University, Institute of Mathematics, Albrechtstraße 28a, D-49076 Osnabrück, Germany |
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Abstract: | The zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated, as the space dimension tends to infinity. The model includes the zero cell of stationary and isotropic Poisson hyperplane tessellations as well as the typical cell of a stationary Poisson Voronoi tessellation as special cases. It is shown that asymptotically in the space dimension, with overwhelming probability these cells satisfy the hyperplane conjecture, if the distance exponent and the intensity parameter are suitably chosen dimension-dependent functions. Also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed. In the background are new identities linking the f-vector of the zero cell to certain dual intrinsic volumes. |
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Keywords: | primary 52A22 52A23 52B05 secondary 60D05 52A39 52C45 |
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