Phase transition for the volume of high‐dimensional random polytopes

The beta polytope Pn,dβ is the convex hull of n i.i.d. random points distributed in the unit ball of ℝd according to a density proportional to (1−‖x‖2)β if β>−1 (in particular, β=0 corresponds to the uniform distribution in the ball), or uniformly on the unit sphere if β=−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 β=0, their number of vertices.