Probability that \(n\) random points are in convex position
type: Item
instance of: scholarly article
title: Probability that \(n\) random points are in convex position
zbMATH Open document ID: 0820.60007
DOI: 10.1007/BF02574070
published in: Discrete \& Computational Geometry
publication date: 1995-07-02T00:00:00Z
full work available at URL: https://eudml.org/doc/131388
review text: Suppose that \(n\) points are chosen independently and uniformly from a parallelogram. The paper shows that the probability that they are the vertices of a convex polygon is \(\left( \left( \begin{smallmatrix} 2n - 2\\ n - 2\end{smallmatrix} \right)/n!\right)^ 2\). The proof proceeds by reduction to the unit square and a combinatorial treatment of approximating grid versions. A note added in proof states the following probability for the analogous problem in a triangle: \(2^ n(3n - 3)!/((n - 1)!)^ 3(2n)!\) (proof to be given elsewhere).
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 764252
zbMATH Keywords: random points
zbMATH Keywords: convex polygon
zbMATH Keywords: combinatorial treatment of approximating grid versions
author: Pavel Valtr
reviewed by: Richard A. Vitale
MaRDI profile type: Publication
cites work: The limit shape of convex lattice polygons
cites work: Random triangles in convex regions
cites work: Q4274036
OpenAlex ID: W2066658205
description: scientific article; zbMATH DE number 764252