Let X 1 , . . . , X N , N > n, be independent random points in R n , distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more general measures of the convex hulls of these random point sets, as the space dimension n tends to infinity. The dual setting of polytopes generated by random halfspaces is also investigated.

Let X 1 , . . . , X n be independent random points that are distributed according to a probability measure on R d and let P n be the random convex hull generated by X 1 , . . . , X n (n ≥ d + 1). Natural classes of probability distributions are characterized for which, by means of Blaschke-Petkantschin formulae from integral geometry, one can show that the mean facet number of P n is strictly monotonically increasing in n.

Short and transparent proofs of central limit theorems for intrinsic volumes of random polytopes in smooth convex bodies are presented. They combine different tools such as estimates for floating bodies with Stein's method from probability theory.

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