Asymptotic Methods for Random Tessellations

  • Pierre CalkaEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2068)


In this chapter, we are interested in two classical examples of random tessellations which are the Poisson hyperplane tessellation and Poisson–Voronoi tessellation. The first section introduces the main definitions, the application of an ergodic theorem and the construction of the so-called typical cell as the natural object for a statistical study of the tessellation. We investigate a few asymptotic properties of the typical cell by estimating the distribution tails of some of its geometric characteristics (inradius, volume, fundamental frequency). In the second section, we focus on the particular situation where the inradius of the typical cell is large. We start with precise distributional properties of the circumscribed radius that we use afterwards to provide quantitative information about the closeness of the cell to a ball. We conclude with limit theorems for the number of hyperfaces when the inradius goes to infinity.


Convex Hull Extreme Point Poisson Point Process Voronoi Tessellation Distribution Tail 
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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Université de RouenRouenFrance

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