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 Fi (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.
Unable to display preview. Download preview PDF.