Variances and central limit theorems for random beta-polytopes and in other geometric models

2025/08/29 by Ferenc Fodor, Fodor, Ferenc, Balázs Grünfelder +1
Mathematics · #Point processes and geometric inequalities #Geometry and complex manifolds #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2508.21392

Abstract

We prove matching asymptotic lower and upper bounds on the variances of the intrinsic volumes and the number of k-faces of d-dimensional random beta-polytopes. Using Stein's methods, we establish central limit theorems for the intrinsic volumes. We also prove asymptotic upper bounds on the variances of the volume and vertex number of spherical random polytopes in spherical convex bodies, and hyperbolic random polytopes in convex bodies in hyperbolic space. Moreover, we consider a circumscribed model on the sphere.

Citations

• Beta Polytopes and Beta Cones: An Exactly Solvable Model in Geometric Probability
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• Variance bounds for disc-polygons
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• Asymptotic normality for random polytopes in non-Euclidean geometries
• Beta polytopes and Poisson polyhedra: f-vectors and angles
• Expected intrinsic volumes and facet numbers of random beta-polytopes
• The floating body in real space forms
• The Spherical Convex Floating Body
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Related

• Expected intrinsic volumes and facet numbers of random beta-polytopes
• Random polytopes: central limit theorems for intrinsic volumes
• Limit theorems for random polytopes with vertices on convex surfaces
• Extremal random beta polytopes
• Beta Polytopes and Beta Cones: An Exactly Solvable Model in Geometric Probability