  Let $X_1,\ldots,X_N$, $N>n$, be independent random points in $\mathbb{R}^n$,
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.
