Phase transition for the volume of high-dimensional random polytopes

Abstract

The beta polytope $P_{n,d}^{\beta }$ is the convex hull of $n$ i.i.d. random points distributed in the unit ball of ${\mathbb{R}}^d$ according to a density proportional to $(1-\Vert x{\Vert }^2{)}^{\beta }$ if $\beta >-1$ (in particular, $\beta =0$ corresponds to the uniform distribution in the ball), or uniformly on the unit sphere if $\beta =-1$. We show that the expected normalized volumes of high-dimensional beta polytopes exhibit a phase transition and we describe its shape. We derive analogous results for the intrinsic volumes of beta polytopes and, when $\beta =0$, their number of vertices.

15 pages, accepted for publications in Random Structures and Algorithms, this revision includes an appendix "Random simplices have small volumes"