A generalization of hadwiger's transversal theorem to intersecting sets

Abstract

Given an ordered family of compact convex sets in the plane, if every three sets can be intersected by some directed line "consistent" with the ordering, then there exists a common transversal of the family. This generalizes Hadwiger's Transversal Theorem to families of compact convex sets which are not necessarily pairwise disjoint. If every six sets can be intersected by some directed line "consistent" with the ordering, then there exists a common transversal which is "consistent" with the ordering. If the family is pairwise disjoint and every four sets can be intersected by some directed line "consistent" with the ordering, then there exists a common transversal which is "consistent" with the ordering. © 1990 Springer-Verlag New York Inc.