Extremes for the inradius in the Poisson line tessellation

TL;DR

This paper studies the distribution of the largest and smallest inradii in a Poisson line tessellation within a large window, revealing their limit distributions and characterizing the shape of cells with minimal inradius.

Contribution

It provides the first limit distribution results for extreme inradii in Poisson line tessellations and characterizes the minimal inradius cell shape as a triangle.

Findings

Limit distributions for the largest and smallest inradii derived.

Cells minimizing inradius are asymptotically triangular.

Poisson approximation used for analysis.

Abstract

A Poisson line tessellation is observed within a window. 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 the window in the limit as the window is scaled to infinity. We additionally prove that the limit shape of the cells minimising the inradius is a triangle.