Phase transition for the volume of high-dimensional random polytopes

The beta polytope (Formula presented.) is the convex hull of n i.i.d. random points distributed in the unit ball of (Formula presented.) according to a density proportional to (Formula presented.) if (Formula presented.) (in particular, (Formula presented.) corresponds to the uniform distribution in the ball), or uniformly on the unit sphere if (Formula presented.). 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 (Formula presented.), their number of vertices.