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Extremes for the inradius in the Poisson line tessellation

  • Nicolas Chenavier (a1) and Ross Hemsley (a2)
Abstract

A Poisson line tessellation is observed in the window Wρ := B(0, π-1/2ρ1/2) for ρ > 0. With each cell of the tessellation, we associate the inradius, which is the radius of the largest ball contained in the cell. Using the Poisson approximation, we compute the limit distributions of the largest and smallest order statistics for the inradii of all cells whose nuclei are contained in Wρ as ρ goes to ∞. We additionally prove that the limit shape of the cells minimising the inradius is a triangle.

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Corresponding author
* Postal address: LMPA Joseph Liouville, Université du Littoral Côte d'Opale, 50 rue Ferdinand Buisson, BP 699, F-62228 Calais Cedex, France. Email address: nicolas.chevavier@lmpa.univ-littoral.fr
** Postal address: Inria, BP 93, 06902 Sophia-Antipolis Cedex, France.
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
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