Lectures on random Voronoi tessellations
type: Item
instance of: scholarly article
title: Lectures on random Voronoi tessellations
zbMATH Open document ID: 0812.60016
author: Jesper Møller
published in: Lecture Notes in Statistics
publication date: 1994-06-14T00:00:00Z
review text: A tessellation (mosaic) is a subdivision of \(\mathbb{R}^ d\) into \(d\)- dimensional non-overlapping sets (cells). The book provides a rigorous mathematical treatment of random Voronoi tessellations. These tessellations arise from spatial point processes by assuming that cells correspond to the points of the process (nuclei) and each cell consists of all points which have the given point of the process as its nearest nucleus. Chapters 1-2 are devoted to non-random properties of Voronoi tessellations related to the geometric structure of the cell conditioned on the nuclei configuration. In particular, Chapter 2 gives necessary background from the integral geometry. Chapter 3 treats arbitrary stationary Voronoi tessellations using Palm measure theory. The principal problem is the study of the geometry of the ``typical'' cell in the tessellation. Chapter 4 concerns the classical Poisson-Voronoi tessellations which appear if the nuclei form a stationary Poisson point process. This assumption allows computations of many interesting theoretical characteristics of the typical cell. However, even in this case many quantities are known only from simulations. The author gives full proofs of the results and assumes no background knowledge of Voronoi tessellations and, in general, of spatial statistics and stochastic geometry. This is the unique book where many probabilistic results on Voronoi tessellations and their proofs can be found.
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 588798
zbMATH Keywords: point process
zbMATH Keywords: mosaic
zbMATH Keywords: tessellation
zbMATH Keywords: spatial point processes
zbMATH Keywords: non-random properties of Voronoi tessellations
zbMATH Keywords: Poisson-Voronoi tessellations
reviewed by: Ilya S. Molchanov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 588798