Note on the variances of random beta-prime polytopes

TL;DR

This paper investigates the variances of intrinsic volumes and face counts of random polytopes generated from beta-prime distributions in high-dimensional space, establishing lower bounds and linking to spherical models.

Contribution

It provides the first variance lower bounds for beta-prime random polytopes and connects these results to spherical polytopes, highlighting their distinct heavy-tailed distribution properties.

Findings

Lower bounds for variances of intrinsic volumes

Lower bounds for variances of the f-vector

Variance results extend to spherical polytopes

Abstract

We consider random polytopes in the $d$-dimensional Euclidean space that are the convex hulls i.i.d. random points selected according to beta-prime distributions. These distributions are rotationally symmetric, heavy-tailed, and their support is the entire space, making them distinct from other commonly studied distributions, for instance, the uniform and Gaussian distributions. We prove lower bounds for the variances of the intrinsic volumes and the $f$-vector of such random polytopes. Beta-prime random polytopes are the push-forwards of spherical random polytopes, which are the convex hulls of random points chosen in the upper open hemisphere according to some rotationally symmetric distribution, including the uniform distribution in the open half-sphere. Our variance lower bounds also transfer to the spherical settings.