Canonical Theorems for Convex Sets
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 ⋃i=1rFi. (Here cr is a positive constant depending only on r.) This generalizes and sharpens some results of Erdős and Szekeres, Bisztriczky and Fejes Tóth, Bárány and Valtr, and others.
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