Threshold phenomena for high-dimensional random polytopes

Let X1,.,XN, N > n, be independent random points in ℝn, 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.