Sylvester's question: The probability that \(n\) points are in convex position
For a convex body \(K\) in the plane, let \(p(n, K)\) denote the probability that \(n\) random points uniformly drawn from \(K\) are in convex position, that is, none of them lies in the convex hull of the others. The author shows that, as \(n\) goes to infinity, \(n^2(p(n, K))^{1/n}\) tends to a finite positive limit. This limit is explicitly determined as \({1\over 4} e^2A^3(K)\), where \(A(K)\) is the supremum of the affine perimeters of all convex sets \(S\subset K\). Further results include the asymptotic for the expected number of non-degenerate convex polygons spanned by all subsets of \(n\) random points in \(K\).
scientific article; zbMATH DE number 1496118