CANONICAL THEOREMS FOR CONVEX-SETS

Let F be a family of pairwise disjoint compact convex sets in the plan e 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 right perpendicular c(r)n left perpendicular-membered subfamilies F-i (1 le ss than or equal to i less than or equal to 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 boolean O Ri=1r F-i. (Here c(r) is a positive constant depending only on r.) Thi s generalizes and sharpens some results of Erdos and Szekeres, Bisztri czky and Fejes Toth, Barany and Valtr, and others.