Canonical theorems for convex sets
type: Item
instance of: scholarly article
title: Canonical theorems for convex sets
zbMATH Open document ID: 0905.52001
DOI: 10.1007/PL00009360
published in: Discrete \& Computational Geometry
publication date: 1999-01-04T00:00:00Z
review text: This beautiful paper discusses a theorem that generalizes the famous Erdős-Szekeres theorem (for every \(r \geq 3\), there exists an integer \(f(r)\) such that any set of at least \(f(r)\) points in the plane has \(r\) elements as vertices of a convex \(r\)-gon). We say that a collection of convex sets are in convex position if none of its members is contained in the convex hull of the union of the others. In the present paper the main result says: For any family \(F\) of \(n\) pairwise disjoint compact sets in the plane, such that none of them is contained in the convex hull of two others, and \(r\) a positive integer there are \(r\) disjoint subfamilies \(F_i\) (\(1 \leq i\geq r\)) of cardinality \(c_rn\) (\(c_r\) is a constant that depends on \(r\)) such that no matter how we pick an element \(p_i\) from each \(F_i\) , they are in convex position (as convex sets). In other words, every \(F_i\) appears on the boundary of the convex hull of \(\bigcup_{i=1}^{r} F_i\). The authors present two other Ramsey-type theorems.
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52C10
Mathematics Subject Classification ID: 05C55
zbMATH DE Number: 1160552
zbMATH Keywords: distance
zbMATH Keywords: Erdős-Szekeres theorem
zbMATH Keywords: convex sets
author: János Pach
author: József Solymosi
MaRDI profile type: Publication
full work available at URL: pl00009360
OpenAlex ID: W2051995105
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description: scientific article; zbMATH DE number 1160552