On the oscillation of the expected number of extreme points of a random set (Q756837)

Let \(X_ 1,...,X_ n\) be i.i.d. r.v.'s with radially symmetric distribution in the plane and let \(N_ n\) be the number of extremal points on the convex hull formed by \(X_ 1,...,X_ n\). It is shown that there exists a radially symmetric distribution (with unimodal density) such that, for arbitrary \(\epsilon >0\) and a given non-decreasing sequence \(\omega_ n\to \infty\), the inequalities \[ E N_ n\geq n/\omega_ n\quad (E N_ n\geq n^{1/3}/\omega_ n)\text{ and } E N_ n\leq 4+\epsilon \] hold for infinitely many n.