, Volume 19, Issue 3, pp 427-435

Canonical Theorems for Convex Sets

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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 ⌊ c r n⌋-membered subfamilies F i (1 ≤ i ≤ r) such that no matter how we pick one element F i from each F i , they are in convex position, i.e., every F i appears on the boundary of the convex hull of ⋃ i=1 r F i . (Here c r 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.