Sylvester’s question: The probability that \(n\) points are in convex position. (English) Zbl 0959.60006

Bárány, Imre

Ann. Probab. 27, No. 4, 2020-2034 (1999).

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 \(K\).

Reviewer: Ilya S.Molchanov (Glasgow)

MSC:

60D05	Geometric probability and stochastic geometry
52A22	Random convex sets and integral geometry (aspects of convex geometry)

Keywords:

affine perimeter; random points in convex position; limit shape