Variances and central limit theorems for random beta-polytopes and in other geometric models
Ferenc Fodor, Balázs Grünfelder
TL;DR
The paper establishes sharp variance rates and central limit theorems for intrinsic volumes and f-vectors of random beta-polytopes in Euclidean, spherical, and hyperbolic geometries. It develops a robust geometric toolkit—floating bodies, the economic cap covering, weighted non-uniform models, and gnomonic projections—to translate non-Euclidean problems into Euclidean analyses and to bound variances via Efron–Stein and Kubota formulas. Central limit theorems are proven for intrinsic volumes with explicit convergence rates in Wasserstein distance using Stein's method, while variance bounds for volume and vertex counts are extended to spherical and hyperbolic circumscribed models and to the inscribed f-vector. The results unify Euclidean and non-Euclidean random-polytope theory, providing matching asymptotics and connecting inscribed and circumscribed frameworks through polarity and projection techniques.
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.