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

## TL;DR

This paper establishes asymptotic bounds and central limit theorems for variances and distributions of geometric features in various random polytopes across Euclidean, spherical, and hyperbolic spaces.

## Contribution

It provides the first comprehensive variance bounds and CLTs for intrinsic volumes and face counts in multiple geometric models, extending previous results.

## Key findings

- Matching asymptotic bounds on variances of intrinsic volumes and face counts.
- Central limit theorems for intrinsic volumes using Stein's method.
- Asymptotic upper bounds on variances of volume and vertices in spherical and hyperbolic polytopes.

## 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.