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October 1999 Sylvester’s Question: The Probability That $n$ Points are in Convex Position
Imre Bárány
Ann. Probab. 27(4): 2020-2034 (October 1999). DOI: 10.1214/aop/1022874826

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.

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Imre Bárány. "Sylvester’s Question: The Probability That $n$ Points are in Convex Position." Ann. Probab. 27 (4) 2020 - 2034, October 1999. https://doi.org/10.1214/aop/1022874826

Information

Published: October 1999
First available in Project Euclid: 31 May 2002

zbMATH: 0959.60006
MathSciNet: MR1742899
Digital Object Identifier: 10.1214/aop/1022874826

Subjects:
Primary: 60D05
Secondary: 52A22

Rights: Copyright © 1999 Institute of Mathematical Statistics

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Vol.27 • No. 4 • October 1999
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