## 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. |

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angle arbitrary assume ball Borel set boundary Brakke centroid contains d-n+k d-polytope defined Delaunay cells Delaunay edge Delaunay tessellation denotes density dyd-n 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 La-k Lemma line segment locally finite mathematical measurable function Models Møller non-negative measurable function Note nuclei nullset number of vertices obtain order moments Palm distribution planar section planar Voronoi point process Poisson point process Poisson-Voronoi cell Proposition 3.2.3 random Voronoi tessellations Remark simulated spatial point process spatial Voronoi Statistical stochastic tessella tion translation invariant typical Poisson-Voronoi typical Voronoi cell Verify vertex void-probabilities Voronoi and Delaunay Voronoi cells Voronoi edges Voronoi tessellation Ε Σ Σ ΕΦ