Sylvester’s Question: The Probability That $n$ Points are in Convex Position

Abstract

For a convex body $K$ in the plane, let $p(n, K)$ denote the probability that $n$ random, independent, and uniform points from $K$ are in convex position, that is, none of them lies in the convex hull of the others. Here we determine the asymptotic behavior of $p(n, K)$ by showing that, as $n$ goes to infinity,$ {n^2}^n\sqrt{p(n,K)}$ tends to a finite and positive limit.