Suppose $X_1, ..., X_n$ are i.i.d. random variables uniformly distributed in the unit ball in $\mathbb{R}^m$. What is the expected number of vertices that their convex hull has?
The only thing I managed to prove here was:
$$P(\text{ convex hull of }X_1, ..., X_n\text{ has exactly }k\text{ vertices}) = C_n^k P(X_{k+1}, ... ,X_n \text{ lie in the convex hull of }X_1, ..., X_n)$$
Not sure, however, whether this helps or not.
The expected number of vertices in the convex hull is $O(n^{\frac{m - 1}{m + 1}})$ for $n \to \infty$. It was proved in 1970 by H. Raynaud in "Sur l’enveloppe convex des nuages de points aleatoires dans $\mathbb{R}_n$".