Phase transition for the volume of high‐dimensional random polytopes. Issue 4 (28th December 2020) - Bonnet, Gilles; Kabluchko, Zakhar; Turchi, Nicola
Abstract: The beta polytope P n, d β is the convex hull of n i.i.d. random points distributed in the unit ball of ℝ d 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.