The limit shape of the zero cell in a stationary Poisson hyperplane tessellation

• Daniel Hug ^[1] ; Matthias Reitzner ^[2] ; Rolf Schneider ^[1]
  1. [1] University of Fribourg

     University of Fribourg

     Friburgo, Suiza

     
  2. [2] Vienna University of Technology

     Vienna University of Technology

     Innere Stadt, Austria

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 32, Nº. 1, 2, 2004, págs. 1140-1167
• Idioma: inglés
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• Resumen

  • In the early 1940s, D. G. Kendall conjectured that the shape of the zero cell of the random tessellation generated by a stationary and isotropic Poisson line process in the plane tends to circularity given that the area of the zero cell tends to ∞. A proof was given by I. N. Kovalenko in 1997. This paper generalizes Kovalenko's result in two directions: to higher dimensions and to not necessarily isotropic stationary Poisson hyperplane processes. In the anisotropic case, the asymptotic shape of the zero cell depends on the direction distribution of the hyperplane process and is obtained from it via an application of Minkowski's existence theorem for convex bodies with given area measures.