## Lectures on random Voronoi tessellations.(English)Zbl 0812.60016

Lecture Notes in Statistics (Springer). 87. New York, NY: Springer- Verlag. 134 p. (1994).
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.

### MSC:

 60D05 Geometric probability and stochastic geometry 60-02 Research exposition (monographs, survey articles) pertaining to probability theory 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)