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.

The first comprehensive treatment of Poisson hyperplane tessellations Considers all aspects of a beautiful and accessible subject Proceeds from elementary foundations to most recent active research

Autorentext

Daniel Hug: Born 1965, Studies of Mathematics and Physics in Freiburg, Diploma 1991, PhD 1994 and Habilitation 2000 in Mathematics (Univ. Freiburg). Assistant Professor at TU Vienna (2000), 2000--2005 Assistant/Associate Professor Univ. Freiburg, 2005--2007 trained and acted as a High School Teacher, 2007 Professor Univ. Duisburg-Essen, 2007--2011 Associate Professor in Karlsruhe, Professor in Karlsruhe (KIT) since 2011. Rolf Schneider: Born 1940, Studies of Mathematics and Physics in Frankfurt/M, Diploma 1964, PhD 1967 (Frankfurt), Habilitation 1969 (Bochum), 1970 Wissenschaftlicher Rat and Professor Univ. Frankfurt, 1970 Professor TU Berlin, 1974 Professor Univ. Freiburg, 2003 Dr. h.c. Univ. Salzburg, 2005 Emeritus.

Inhalt

• 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.