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 ⌊ 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.