Separating convex sets in the plane

Abstract

Given a setA inR2 and a collectionS of plane sets, we say that a lineL separatesA fromS ifA is contained in one of the closed half-planes defined byL, while every set inS is contained in the complementary closed half-plane. We prove that, for any collectionF ofn disjoint disks inR2, there is a lineL that separates a disk inF from a subcollection ofF with at least ?(n?7)/4? disks. We produce configurationsHn andGn, withn and 2n disks, respectively, such that no pair of disks inHn can be simultaneously separated from any set with more than one disk ofHn, and no disk inGn can be separated from any subset ofGn with more thann disks. We also present a setJm with 3m line segments inR2, such that no segment inJm can be separated from a subset ofJm with more thanm+1 elements. This disproves a conjecture by N. Alonet al. Finally we show that ifF is a set ofn disjoint line segments in the plane such that they can be extended to be disjoint semilines, then there is a lineL that separates one of the segments from at least ?n/3?+1 elements ofF.