Lower bounds to Helly numbers of line transversals to disjoint congruent balls

Abstract

A line l is a transversal to a family F of convex objects in a"e (d) if it intersects every member of F. In this paper we show that for every integer d >= 3 there exists a family of 2d-1 pairwise disjoint unit balls in a"e (d) with the property that every subfamily of size 2d - 2 admits a transversal, yet any line misses at least one member of the family. This answers a question of Danzer from 1957. Crucial to the proof is the notion of a pinned transversal, which means an isolated point in the space of transversals. Here we investigate minimal pinning configurations and construct a family F of 2d-1 disjoint unit balls in R-d with the following properties: (i) The space of transversals to F is a single point and (ii) the space of transversals to any proper subfamily of F is a connected set with non-empty interior.