Weak convergence of the intersection point process of Poisson hyperplanes

TL;DR

This paper proves that the intersection points of a Poisson hyperplane process, when scaled appropriately, converge to a Poisson process with a power-law intensity, with implications for geometric structures.

Contribution

It establishes the weak convergence of the intersection point process of Poisson hyperplanes to a power-law Poisson process and corrects a previous conjecture in computational geometry.

Findings

Convergence of the scaled intersection point process to a power-law Poisson process.

Provided bounds on the convergence speed using Kantorovich-Rubinstein distance.

Disproved and corrected a conjecture regarding the convex hull and f-vector convergence.

Abstract

This paper deals with the intersection point process of a stationary and isotropic Poisson hyperplane process in $\mathbb{R}^d$ of intensity $t>0$, where only hyperplanes that intersect a centred ball of radius $R>0$ are considered. Taking $R=t^{-\frac{d}{d+1}}$ it is shown that this point process converges in distribution, as $t\to\infty$, to a Poisson point process on $\mathbb{R}^d\setminus\{0\}$ whose intensity measure has power-law density proportional to $\|x\|^{-(d+1)}$ with respect to the Lebesgue measure. A bound on the speed of convergence in terms of the Kantorovich-Rubinstein distance is provided as well. In the background is a general functional Poisson approximation theorem on abstract Poisson spaces. Implications on the weak convergence of the convex hull of the intersection point process and the convergence of its $f$-vector are also discussed, disproving and correcting thereby a conjecture of Devroye and Toussaint [J.\ Algorithms 14.3 (1993), 381--394] in computational geometry.