## Extremes for the inradius in the Poisson line tessellation.(English)Zbl 1342.60011

Summary: A Poisson line tessellation is observed in the window $$\boldsymbol{W}_\rho:=B(0,\pi^{-1/2}\rho^{1/2})$$ for $$\rho>0$$. 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 $$\boldsymbol{W}_\rho$$ as $$\rho$$ goes to $$\infty$$. We additionally prove that the limit shape of the cells minimising the inradius is a triangle.

### MSC:

 60D05 Geometric probability and stochastic geometry 60G70 Extreme value theory; extremal stochastic processes 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60F05 Central limit and other weak theorems 62G32 Statistics of extreme values; tail inference
Full Text: