Poisson Hyperplane Tessellations

This book is the first comprehensive presentation of a central topic of stochastic geometry: random mosaics that are generated by Poisson processes of hyperplanes. It thus connects a basic notion from probability theory, Poisson processes, with a fundamental object of geometry. The independence properties of Poisson processes and the long-range influence of hyperplanes lead to a wide range of phenomena which are of interest from both a geometric and a probabilistic point of view. A Poisson hyperplane tessellation generates many random polytopes, also a much-studied object of stochastic geometry. The book offers a variety of different perspectives and covers in detail all aspects studied in the original literature. The work will be useful to graduate students (advanced students in a Master program, PhD students), and professional mathematicians. The book can also serve as a reference for researchers in fields of physics, computer science, economics or engineering.

- 1 Notation. - 2 Hyperplane and particle processes. - 3 Distribution-independent density relations. - 4 Poisson hyperplane processes. - 5 Auxiliary functionals and bodies. - 6 Zero cell and typical cell. - 7 Mixing and ergodicity. - 8 Observations inside a window. - 9 Central limit theorems. - 10 Palm distributions and related constructions. - 11 Typical faces and weighted faces. - 12 Large cells and faces. - 13 Cells with a given number of facets. - 14 Small cells. - 15 The K-cell under increasing intensities. - 16 Isotropic zero cells. - 17 Functionals of Poisson processes and applications. - 18 Appendix: Some auxiliary results.