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Volume degeneracy of the typical cell and the chord length distribution for Poisson-Voronoi tessellations in high dimensions

Alishahi, K ; Sharif University of Technology | 2008

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  1. Type of Document: Article
  2. DOI: 10.1239/aap/1231340158
  3. Publisher: 2008
  4. Abstract:
  5. This paper is devoted to the study of some asymptotic behaviors of Poisson-Voronoi tessellation in the Euclidean space as the space dimension tends to ∞. We consider a family of homogeneous Poisson-Voronoi tessellations with constant intensity λ in Euclidean spaces of dimensions n = 1, 2, 3,... First we use the Blaschke-Pètkantschin formula to prove that the variance of the volume of the typical cell tends to 0 exponentially in dimension. It is also shown that the volume of intersection of the typical cell with the co-centered ball of volume u converges in distribution to the constant λ-1 (1-e-λu). Next we consider the linear contact distribution function of the Poisson-Voronoi tessellation and compute the limit when the space dimension goes to ∞. As a by-product, the chord length distribution and the geometric covariogram of the typical cell are obtained in the limit. © Applied Probability Trust 2008
  6. Keywords:
  7. Distribution functions ; Particle size analysis ; Poisson equation ; Blaschke-Petkantschin formula ; Chord length distribution ; Geometric covariogram ; High dimension ; Linear contact distribution ; Poisson-Voronoi tessellation ; Typical cell ; Poisson distribution
  8. Source: Advances in Applied Probability ; Volume 40, Issue 4 , July , 2008 , Pages 919-938 ; 00018678 (ISSN)
  9. URL: https://www.cambridge.org/core/journals/advances-in-applied-probability/article/volume-degeneracy-of-the-typical-cell-and-the-chord-length-distribution-for-poissonvoronoi-tessellations-in-high-dimensions/105375C626BD96DBC1BF71A546AF97D3