Sylvester's question: The probability that \(n\) points are in convex position
type: Item
instance of: scholarly article
title: Sylvester's question: The probability that \(n\) points are in convex position
zbMATH Open document ID: 0959.60006
author: Imre Bárány
published in: The Annals of Probability
publication date: 2001-04-26T00:00:00Z
review text: 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 \(K\).
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 52A22
zbMATH DE Number: 1496118
zbMATH Keywords: affine perimeter
zbMATH Keywords: random points in convex position
zbMATH Keywords: limit shape
reviewed by: Ilya S. Molchanov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1022677559
OpenAlex ID: W2389044384
DOI: 10.1214/AOP/1022677559
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description: scientific article; zbMATH DE number 1496118