Canonical theorems for convex sets

Abstract

Let F be a family of pairwise disjoint compact convex sets in the plane such that none of them is contained in the convex hull of two others, and let r be a positive integer. We show that F has r disjoint ⌊crn⌋-membered subfamilies Fi (1 ≤ i ≤ r) such that no matter how we pick one element Fi from each Fi, they are in convex position, i.e., every Fi appears on the boundary of the convex hull of ∪ri=1 Fi. (Here cr is a positive constant depending only on r.) This generalizes and sharpens some results of Erdos and Szekeres, Bisztriczky and Fejes Tóth, Bárány and Valtr, and others.