Extremes for the inradius in the Poisson line tessellation

Abstract

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.The Poisson line tessellation LetX be a stationary and isotropic Poisson line process of intensitŷ γ = π in R 2 endowed with its scalar product ·, · and its Euclidean norm | · |. By A, we shall denote the set of affine lines which do not pass through the origin 0 ∈ R 2 . Each line can be written asfor some t ∈ R, u ∈ S, where S is the unit sphere in R 2 . When t > 0, this representation is unique. The intensity measure ofX is then given byfor all Borel subsets E ⊆ A, where A is endowed with the Fell topology (see for example Schneider and Weil [21], p563) and where σ(·) denotes the uniform measure on S with the normalisation σ(S) = 2π. The set of closures of the connected components of R 2 \X defines a stationary and isotropic random tessellation with intensity γ (2) = π (see for example (10.46) in Schneider and Weil [21]) which is the so-called Poisson line tessellation, m pht . By a slight abuse of notation, we also writeX to denote the union of lines. An example of the Poisson line tessellation in R 2 is depicted in Figure 1. Let B(z, r) denote the (closed) disc of radius r ∈ R + , centred at z ∈ R 2 and let K be the family of convex bodies (i.e. convex compact sets in R 2 with non-empty interior), endowed with the Hausdorff topology. With each convex body K ∈ K, we may now define the inradius,When there exists a unique z ∈ R 2 such that B(z , r(K)) ⊂ K, we define z(C) := z to be the incentre of K. If no such z exists, we take z(K) := 0 ∈ R 2 . Note that each cell C ∈ m pht has a unique z

The largest order statistics for the inradius in an isotropic STIT tessellation

“…We now compare our method to the one used in [4]. As mentioned on p. 3, our result is similar to Theorem 1.1 (ii) of [4].…”

“…We now compare our method to the one used in [4]. As mentioned on p. 3, our result is similar to Theorem 1.1 (ii) of [4]. However, the proof of this theorem is based on the method of moments, which leads to very technical computations of combinatorics, and does not provide rate of convergence for the Poisson approximation.…”

“…Lemmas concerning an upper bound for b 2 The arguments showing that b 2 converges to 0 are mainly inspired by [4], and they rely on a "local condition", i.e. on an upper bound for the probability that the incircles of two cells with centers in a short distance simultaneously exceed the threshold v ρ .…”

“…Proof of Lemma 3.2 Let H 1 ∈ H be fixed, with normal direction u 1 ∈ S 1 . According to (2.1), we have Proof of Lemma 3.3 This will be sketched only because it relies on the same idea as the proof of Lemma 3.2, see also the proof of Lemma A.1, (ii) of [4]. Let H 1 , H 2 ∈ H be fixed.…”

Extremal behavior of large cells in the Poisson hyperplane mosaic

“…This result is the answer to the Kendall problem (formulated for the zero cell of a Poisson hyperplane mosaic) and is adapted to the typical cell of the mosaic in [7]. In [4] cells with large (and small) inradius and with center in a window are considered. For dimension d = 2 it is shown that the limit distribution of the largest and smallest order statistics for the inradii converges to a Poisson distribution when the size of the window goes to infinity.…”