Random inscribed polytopes in projective geometries

Abstract

We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are obtained. We deduce these results by proving a general central limit theorem for the weighted volume of the convex hull of random points chosen from the boundary of a smooth convex body according to a positive and continuous density in Euclidean space. In the background are geometric estimates for weighted surface bodies and a Berry–Esseen bound for functionals of independent random variables.

Fluctuations of $λ$-geodesic Poisson hyperplanes in hyperbolic space

“…However, recently there has been a growing interest in random geometric systems in non-Euclidean spaces, most prominently, the spaces of constant negative curvature −1. Examples include the study of random hyperbolic Voronoi tessellations [3,17,16], hyperbolic random polytopes [4,5], the hyperbolic Boolean model [2,36,37], hyperbolic Poisson cylinder processes [8] or hyperbolic random geometric graphs [14,26]. Most closely related to the present paper are the investigations in [19] about Poisson processes of hyperplanes, that is, totally geodesic (d−1)-dimensional submanifolds, in d-dimensional hyperbolic spaces.…”

“…λ-geodesic hyperplanes. Let us now provide a similar non-rigorous analysis of the fluctuations of the random variable S (λ) R defined in (4). We restrict ourselves to the case λ ∈ [0, 1).…”

Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes

“…Originally introduced in [30] and further developed in many subsequent works it has turned out to be a versatile device with a vast of potential applications. As concrete examples we mention the works [8,20,21,22,23,31,32,37] on various models for geometric random graphs, the paper [7] dealing with geometric random simplicial complexes, the application to the classical Boolean model [17], the works [8,16,25,31] dealing with Poisson hyperplane tessellations in Euclidean and non-Euclidean spaces, the applications in [22,36] to Poisson-Voronoi tessellations, the works on excursion sets of Poisson shot-noise processes [19,21] as well as the papers [4,5,22,40,41,42] considering different models for random polytopes. For an illustrative overview on the Malliavin-Stein method for functionals of Poisson processes we refer to the collection of surveys in [29].…”

Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes

“…Originally introduced in [28] and further developed in many subsequent works it has turned out to be a versatile device with a vast of potential applications. As concrete examples we mention the works [8,18,19,20,21,29,30,35] on various models for geometric random graphs, the paper [7] dealing with geometric random simplicial complexes, the application to the classical Boolean model [16], the works [8,15,23,29] dealing with Poisson hyperplane tessellations in Euclidean and non-Euclidean spaces, the applications in [20,34] to Poisson-Voronoi tessellations, the works on excursion sets of Poisson shot-noise processes [17,19] as well as the papers [4,5,20,38,39,40] considering different models for random polytopes. For an illustrative overview on the Malliavin-Stein method for functionals of Poisson processes we refer to the collection of surveys in [27].…”