Lower Bounds for Pinning Lines by Balls (Extended Abstract)

It is known that if n ⩾ 2 d pairwise disjoint balls in R d have a unique line ℓ intersecting them in a given order ≺, one can always remove a ball so that ℓ remains the only line intersecting the balls in the order induced by ≺. We show that the constant 2d is best possible, in any dimension, and derive lower bounds on Helly numbers for sets of line transversals to disjoint balls in arbitrary dimension.