Threshold phenomena for high-dimensional random polytopes

TL;DR

This paper investigates threshold phenomena in the geometric properties of high-dimensional random polytopes generated by independent beta or beta-prime distributed points, as the dimension grows large.

Contribution

It establishes threshold behaviors for volume and intrinsic volumes of convex hulls of high-dimensional random points, including dual polytopes from random halfspaces.

Findings

Threshold phenomena for volume and intrinsic volumes identified

Results apply to both beta and beta-prime distributions

Dual setting of random halfspace-generated polytopes analyzed

Abstract

Let $X_1,\ldots,X_N$, $N>n$, be independent random points in $\mathbb{R}^n$, distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more general measures of the convex hulls of these random point sets, as the space dimension $n$ tends to infinity. The dual setting of polytopes generated by random halfspaces is also investigated.