Limit theorems for the convex hull of random points in higher dimensions

We give a central limit theorem for the number Nn of vertices of the convex hull of n independent and identically distributed random vectors, being sampled from a certain class of spherically symmetric distributions in Rd (d > 1), that includes the normal family. Furthermore, we prove that, among these distributions, the variance of Nn exhibits the same order of magnitude as the expectation as n→∞. The main tools are Poisson approximation of the point process of vertices of the convex hull and (sub/super)-martingales.