On the shape of the convex hull of random points (Q1099869)

Let Prob(d,n) denote the probability that a uniform and independent choice of n points from the unit ball in Euclidean d-space gives a convex hull with exactly n vertices. This paper studies the behaviour of Prob(d,n(d)), as \(d\to \infty\), for certain functions n(d). \textit{C. Buchta} [Monatsh. Math. 102, 91-102 (1986; Zbl 0594.52006)] proved that for \(n=(3/2-\epsilon)d\), \(\epsilon >0\), the probability tends to one. Here the authors show that \[ for\quad n=c 2^{d/2},\quad \lim_{d\to \infty}\Pr ob(d,n)>1-c^ 2, \] \[ and\quad for\quad n=c d^{3/4} 2^{d/2},\quad \lim_{d\to \infty}\Pr ob(d,n)<2e^{-c/2}. \] They also investigate the question whether the convex hull is a neighborly polytope.