Lectures on Random Voronoi TessellationsTessellations 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. |
Contents
| 1 | |
Geometrical properties and other background material | 15 |
Stationary Voronoi tessellations | 43 |
PoissonVoronoi tessellations | 83 |
| 125 | |
Notation index | 132 |
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Common terms and phrases
angle arbitrary ball Borel set boundary C(X₁ centroid contains convex d-n+k d-polytope defined Delaunay cells Delaunay edge Delaunay tessellation denotes density equivariant Exercise facets Figure formula Fubini's theorem Gabriel neighbours geometry Hadwiger's theorem hard core homogeneous Poisson process implies integral geometry intensity intersection invariant under translations isotropic k-dimensional k-facets k-flat Lebesgue measure Lemma locally finite measurable function Models Møller non-negative measurable function nuclei nullset obtain Palm distribution Palm measure planar section planar Voronoi point process Poisson point process Poisson-Voronoi cell polytope Proposition 2.1.7 Proposition 3.2.3 Remark simulated spatial Voronoi stationary Statistical stochastic stochastic geometry tessella tion topological interiors translation invariant typical Poisson-Voronoi typical Voronoi cell vertex void-probabilities Voronoi and Delaunay Voronoi edges Voronoi tessellation X₁ ΕΣ ΕΦ λκ λο Σ Σ Τ Φ ΧΕ Φ ΧΕΦΚ