Threshold phenomena for high-dimensional random polytopes

Abstract

Let $X_1,\dots ,X_N$, $N>n$, 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.

26 pages