The Typical Cell of a Voronoi Tessellation on the Sphere

Kabluchko, Zakhar; Thäle, Christoph

doi:10.1007/s00454-021-00315-2

Cited by 11 publications

(5 citation statements)

References 30 publications

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“…Related polytopes. The results listed above complement our findings for beta and beta' polytopes in [30,31,33,34,36,37], which in turn have found application to Voronoi and hyperplane tessellations in Euclidean and spherical spaces. The beta polytopes are defined as convex hulls of i.i.d.…”

Section: Introductionsupporting

confidence: 83%

Beta-star polytopes and hyperbolic stochastic geometry

Godland¹,

Kabluchko²,

Thäle³

2021

Preprint

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Motivated by problems of hyperbolic stochastic geometry we introduce and study the class of beta-star polytopes. A beta-star polytope is defined as the convex hull of an inhomogeneous Poisson processes on the complement of the unit ball in R d with density proportional to ( x 2 − 1) −β , where x > 1 and β > d/2. Explicit formulas for various geometric and combinatorial functionals associated with beta-star polytopes are provided, including the expected number of k-dimensional faces, the expected external angle sums and the expected intrinsic volumes. Beta-star polytopes are relevant in the context of hyperbolic stochastic geometry, since they are tightly connected to the typical cell of a Poisson-Voronoi tessellation as well as the zero cell of a Poisson hyperplane tessellation in hyperbolic space. The general results for beta-star polytopes are used to provide explicit formulas for the expected fvector of the typical hyperbolic Poisson-Voronoi cell and the hyperbolic Poisson zero cell. Their asymptotics for large intensities and their monotonicity behaviour is discussed as well. Finally, stochastic geometry in the de Sitter half-space is studied as the hyperbolic analogue to recent investigations about random cones generated by random points on half-spheres in spherical or conical stochastic geometry.

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Section: Introductionsupporting

confidence: 83%

Beta-star polytopes and hyperbolic stochastic geometry

Godland¹,

Kabluchko²,

Thäle³

2021

Preprint

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“…This continues a recent trend in stochastic geometry of generalizing known results to the non-Euclidean setting, and in particular to spherical and hyperbolic geometry, see e.g. [5,6,24,28,29,[32][33][34][35][36][37][38].…”

Section: Introductionmentioning

confidence: 79%

Random inscribed polytopes in projective geometries

2021

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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.

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“…The aim of this branch of stochastic geometry is to distinguish those properties of a random geometric system which are universal to some extent from the ones which are sensitive to the underlying geometry, especially to the curvature of the underlying space. We mention by way of example the studies [6,7,20] on random convex hulls, the papers [5,21,22,26,27,29,30,32] on random tessellations as well as the works [4,8,15,16,17,18,40] on geometric random graphs and networks. The present paper continues this line of research and naturally connects to the articles [26,32].…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Intersections of Poisson $ k $-flats in constant curvature spaces

Betken¹,

Hug²,

Thäle³

2023

Preprint

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Poisson processes in the space of k-dimensional totally geodesic subspaces (k-flats) in a ddimensional standard space of constant curvature κ ∈ {−1, 0, 1} are studied, whose distributions are invariant under the isometries of the space. We consider the intersection processes of order m together with their (d − m(d − k))-dimensional Hausdorff measure within a geodesic ball of radius r. Asymptotic normality for fixed r is shown as the intensity of the underlying Poisson process tends to infinity for all m satisfying d − m(d − k) ≥ 0. For κ ∈ {−1, 0} the problem is also approached in the set-up where the intensity is fixed and r tends to infinity. Again, if 2k ≤ d + 1 a central limit theorem is shown for all possible values of m. However, while for κ = 0 asymptotic normality still holds if 2k > d + 1, we prove for κ = −1 convergence to a non-Gaussian infinitely divisible limit distribution in the special case m = 1. The proof of asymptotic normality is based on the analysis of variances and general bounds available from the Malliavin-Stein method. We also show for general κ ∈ {−1, 0, 1} that, roughly speaking, the variances within a general observation window W are maximal if and only if W is a geodesic ball having the same volume as W . Along the way we derive a new integral-geometric formula of Blaschke-Petkantschin type in a standard space of constant curvature.

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The Typical Cell of a Voronoi Tessellation on the Sphere

Cited by 11 publications

References 30 publications

Beta-star polytopes and hyperbolic stochastic geometry

Beta-star polytopes and hyperbolic stochastic geometry

Random inscribed polytopes in projective geometries

Intersections of Poisson $ k $-flats in constant curvature spaces

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