On the variance of the number of extreme points of a random convex hull (Q1962184)

This paper constructs explicit distributions on \(R^2\) such that when \(n\) i.i.d. points are chosen from this distribution, and \(N_n\) is the number of vertices of their convex hull, the variance of \(N_n\) is particularly large. Being a positive variable bounded by \(n\), \(N_n\) must have variance smaller than \(n^2/4\). This paper shows that \(n^2/4-2n-4\) is achievable for every \(n\geq 4\). More significantly, for any increasing sequence \(w(n)\) which goes to infinity, one can construct a single distribution \(P_w\) which makes the variance of \(N_n\) at least \(n^2/w(n)\) infinitely often.