Phase transition for the volume of highdimensional 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 $(1\lVert{x}\rVert^2)^{\beta}$ 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 highdimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.12696
 Bibcode:
 2019arXiv191112696B
 Keywords:

 Mathematics  Probability;
 Mathematics  Metric Geometry;
 52A23 (Primary) 52A22;
 52B11;
 60D05 (Secondary)
 EPrint:
 12 pages