4 Helly-type Theorems and Geometric Transversals

INTRODUCTION Let F be a family of convex sets in R. A geometric transversal is an affine subspace that intersects every member of F . More specifically, for a given integer 0 ≤ k < d, a k-dimensional affine subspace that intersects every member of F is called a ktransversal to F . Typically, we are interested in necessary and sufficient conditions that guarantee the existence of a k-transversal to a family of convex sets in R, and furthermore, to describe the space of all k-transversals to the given family. Not much is known for general k and d, and results deal mostly with the cases k = 0, 1, or d− 1. Helly’s theorem gives necessary and sufficient conditions for the members of a family of convex sets to have a point in common, or in other words, a 0-transversal. Section 4.1 is devoted to some of the generalizations and variations related to Helly’s theorem. In the study of k-transversals, there is a clear distinction between the cases k = 0 and k > 0, and Section 4.2 is devoted to results and questions dealing with the latter case.