Advances in Applied Probability

Extremes for the inradius in the Poisson line tessellation

Nicolas Chenavier and Ross Hemsley

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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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Adv. in Appl. Probab., Volume 48, Number 2 (2016), 544-573.

First available in Project Euclid: 9 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G70: Extreme value theory; extremal processes 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems 62G32: Statistics of extreme values; tail inference

Line tessellation Poisson point process extreme value order statistic


Chenavier, Nicolas; Hemsley, Ross. Extremes for the inradius in the Poisson line tessellation. Adv. in Appl. Probab. 48 (2016), no. 2, 544--573.

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