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Extremes for the inradius in the Poisson line tessellation

Published online by Cambridge University Press:  10 June 2016

Nicolas Chenavier*
Affiliation:
Université du Littoral Côte d'Opale
Ross Hemsley*
Affiliation:
Inria
*
* Postal address: LMPA Joseph Liouville, Université du Littoral Côte d'Opale, 50 rue Ferdinand Buisson, BP 699, F-62228 Calais Cedex, France. Email address: [email protected]
** Postal address: Inria, BP 93, 06902 Sophia-Antipolis Cedex, France.

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.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2016 

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