Extremes for the inradius in the Poisson line tessellation

Nicolas Chenavier (1) , Ross Hemsley (2)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
2 GEOMETRICA - Geometric computing

A Poisson line tessellation is observed in the window Wρ := B(0, π −1/2 ρ 1/2), 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 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 infinity. We additionally prove that the limit shape of the cells minimising the inradius is a triangle.

Mots clés: line tessellations; Poisson point process; extreme values; order statistics

Domaines: Probabilités [math.PR]

Nicolas Chenavier, Ross Hemsley. Extremes for the inradius in the Poisson line tessellation. Advances in Applied Probability, 2016, 48 (2), pp.544-573. ⟨hal-01568858⟩