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Aggregate and fractal tessellations
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  • Published: October 2001

Aggregate and fractal tessellations

  • Konstantin Tchoumatchenko1 &
  • Sergei Zuyev2 

Probability Theory and Related Fields volume 121, pages 198–218 (2001)Cite this article

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  • 17 Citations

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Abstract.

Consider a sequence of stationary tessellations {Θn}, n=0,1,…, of ℝd consisting of cells {C n(x i n)}with the nuclei {x i n}. An aggregate cell of level one, C 0 1(x i 0), is the result of merging the cells of Θ1 whose nuclei lie in C 0(x i 0). An aggregate tessellation Θ0 n consists of the aggregate cells of level n, C 0 n(x i 0), defined recursively by merging those cells of Θn whose nuclei lie in C n −1(x i 0).

We find an expression for the probability for a point to belong to atypical aggregate cell, and obtain bounds for the rate of itsexpansion. We give necessary conditions for the limittessellation to exist as n→∞ and provide upperbounds for the Hausdorff dimension of its fractal boundary and forthe spherical contact distribution function in the case ofPoisson-Voronoi tessellations {Θn}.

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Authors and Affiliations

  1. France Telecom R&D, 38-40 rue du Général Leclerc, 92794 Issy-Moulineaux, France. e-mail: Konstantin.Tchoumatchenko@rd.francetelecom.com, , , , , , FR

    Konstantin Tchoumatchenko

  2. Statistics and Modelling Science Department, University of Strathclyde, 26 Richmond str., Glasgow, G1 1XH, UK. e-mail: sergei@stams.strath.ac.uk, , , , , , GB

    Sergei Zuyev

Authors
  1. Konstantin Tchoumatchenko
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  2. Sergei Zuyev
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Received: 3 June 1999 / Revised version: 22 November 2000 / Published online: 24 July 2001

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Tchoumatchenko, K., Zuyev, S. Aggregate and fractal tessellations. Probab Theory Relat Fields 121, 198–218 (2001). https://doi.org/10.1007/PL00008802

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  • Issue Date: October 2001

  • DOI: https://doi.org/10.1007/PL00008802

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Keywords

  • Distribution Function
  • Hausdorff Dimension
  • Aggregate Cell
  • Fractal Boundary
  • Spherical Contact
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