Canonical Theorems for Convex Sets
- Cite this article as:
- Pach, J. & Solymosi, J. Discrete Comput Geom (1998) 19: 427. doi:10.1007/PL00009360
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.