Lectures on Random Voronoi Tessellations
Tessellations are subdivisions of d-dimensional space into non-overlapping "cells". Voronoi tessellations are produced by first considering a set of points (known as nuclei) in d-space, and then defining cells as the set of points which are closest to each nuclei. A random Voronoi tessellation is produced by supposing that the location of each nuclei is determined by some random process. They provide models for many natural phenomena as diverse as the growth of crystals, the territories of animals, the development of regional market areas, and in subjects such as computational geometry and astrophysics. This volume provides an introduction to random Voronoi tessellations by presenting a survey of the main known results and the directions in which research is proceeding. Throughout the volume, mathematical and rigorous proofs are given making this essentially a self-contained account in which no background knowledge of the subject is assumed.
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a-nik angle arbitrary assume ball Borel set boundary bounded Borel set c(xi c(xol centroid constitute a tessellation contains d-n+k d-nik d-polytope defined Delaunay cells Delaunay edge Delaunay tessellation denotes density EL(e equivariant Exercise facets Figure Fn(x formula 2.2 Fubini's theorem Gabriel neighbours geometry Hadwiger's theorem homogeneous Poisson process implies integral geometry intensity intersection invariant under translations isotropic k-dimensional k-facets k-flat Lebesgue measure Lemma line segment locally finite measurable function measure theory Møller non-negative measurable function nuclei nullset obtain order moments Palm distribution Palm measure planar section point process Poisson point process Poisson-Voronoi cell Poisson-Voronoi tessel process on Rd Quine and Watson Remark sets B C Rd simulated spatial point process stochastic Stoyan tessella tion topological interiors translation invariant typical Poisson-Voronoi typical Voronoi cell Verify vertex void-probabilities Voronoi and Delaunay Voronoi edges Voronoi tessellation