A general study of extremes of stationary tessellations with examples

Let m be a random tessellation in R d , d ≥ 1 , observed in a bounded Borel subset W and f ( ⋅ ) be a measurable function defined on the set of convex bodies. A point z ( C ) , called the nucleus of C , is associated with each cell C of m . Applying f ( ⋅ ) to all the cells of m , we investigate the order statistics of f ( C ) over all cells C ∈ m with nucleus in W ρ = ρ 1 / d W when ρ goes to infinity. Under a strong mixing property and a local condition on m and f ( ⋅ ) , we show a general theorem which reduces the study of the order statistics to the random variable f ( C ) , where C is the typical cell of m . The proof is deduced from a Poisson approximation on a dependency graph via the Chen–Stein method. We obtain that the point process { ( ρ − 1 / d z ( C ) , a ρ − 1 ( f ( C ) − b ρ ) ) , C ∈ m , z ( C ) ∈ W ρ } , where a ρ > 0 and b ρ are two suitable functions depending on ρ , converges to a non-homogeneous Poisson point process. Several applications of the general theorem are derived in the particular setting of Poisson–Voronoi and Poisson–Delaunay tessellations and for different functions f ( ⋅ ) such as the inradius, the circumradius, the area, the volume of the Voronoi flower and the distance to the farthest neighbor.