
Contents
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5.1 Definitions and basic properties of random tessellations 5.1 Definitions and basic properties of random tessellations
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5.1.1 Introduction 5.1.1 Introduction
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5.1.2 Zero‐cell, ergodic means and typical cell 5.1.2 Zero‐cell, ergodic means and typical cell
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5.1.2.1 Zero‐cell 5.1.2.1 Zero‐cell
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5.1.2.2 Ergodic means 5.1.2.2 Ergodic means
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5.1.2.3 Typical cell and Palm measure 5.1.2.3 Typical cell and Palm measure
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5.1.2.4 Realizations of the typical cell 5.1.2.4 Realizations of the typical cell
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5.2 Exact distributional results 5.2 Exact distributional results
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5.2.1 Number of hyperfaces and distribution of the cell conditioned on the number of hyperfaces 5.2.1 Number of hyperfaces and distribution of the cell conditioned on the number of hyperfaces
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5.2.2 Typical k‐face of a section of a Poisson‐Voronoi tessellation 5.2.2 Typical k‐face of a section of a Poisson‐Voronoi tessellation
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5.2.3 The circumscribed radius 5.2.3 The circumscribed radius
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5.3 Asymptotic results 5.3 Asymptotic results
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5.3.1 D. G. Kendall's conjecture 5.3.1 D. G. Kendall's conjecture
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5.3.1.1 Context and useful functionals 5.3.1.1 Context and useful functionals
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5.3.1.2 Estimates of conditional probabilities 5.3.1.2 Estimates of conditional probabilities
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5.3.1.3 Results and examples 5.3.1.3 Results and examples
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5.3.2 Cells with a large inradius 5.3.2 Cells with a large inradius
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5.4 Iterated tessellations 5.4 Iterated tessellations
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5.4.1 Tessellations stable with respect to iteration 5.4.1 Tessellations stable with respect to iteration
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5.4.1.1 Construction of a crack STIT tessellation in a window 5.4.1.1 Construction of a crack STIT tessellation in a window
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5.4.1.2 Extension to a crack STIT tessellation of ℝd and stability with respect to iteration 5.4.1.2 Extension to a crack STIT tessellation of ℝd and stability with respect to iteration
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5.4.2 Iterated tessellations in telecommunications 5.4.2 Iterated tessellations in telecommunications
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References References
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Cite
Abstract
Random tessellations and cellular structures occur in many domains of application, such as astrophysics, ecology, telecommunications, biochemistry and naturally cellular biology (see Stoyan, Kendall and Mecke 1987 or Okabe, Boots, Sugihara and Chiu 2000 for complete surveys). The theoretical study of these objects was initiated in the second half of the twentieth century by D. G. Kendall, J. L. Meijering, E. N. Gilbert and R. E. Miles, notably. Two isotropic and stationary models have emerged as the most basic and useful: the Poisson hyperplane tessellation and the Poisson–Voronoi tessellation. Since then, a large majority of questions raised about random tessellations have concerned statistics of the population of cells (‘how many cells are triangles in the plane?’, ‘how many cells have a volume greater than one?’) or properties of a specific cell (typically the one containing the origin). Two types of results are presented below: exact distributional calculations and asymptotic estimations. In the first part, we describe the two basic constructions of random tessellations (i.e. by throwing random hyperplanes or by constructing Voronoi cells around random nuclei) and we introduce the fundamental notion of typical cell of a stationary tessellation. The second part is devoted to the presentation of exact distributional results on basic geometrical characteristics (number of hyperfaces, typical k‐face, etc.). The following part concerns asymptotic properties of the cells. It concentrates in particular on the well‐known D. G. Kendall conjecture which states that large planar cells in a Poisson line tessellation are close to the circular shape. In the last part, we present some recent models of iterated tessellations which appear naturally in applied fields (study of crack structures, telecommunications). Intentionally, this chapter does not contain an exhaustive presentation of all the models of random tessellations existing in the literature (in particular, dynamical constructions such as Johnson‐Mehl tessellations will be omitted). The aim of the text below is to provide a selective view of recent selected methods and results on a few specific models.
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