Extremes for the inradius in the Poisson line tessellation

Abstract

A Poisson line tessellation is observed in the window W_ρ := B(0, π^-1/2ρ^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 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 W_ρ as ρ goes to ∞. We additionally prove that the limit shape of the cells minimising the inradius is a triangle.