Helly-type theorems for line transversals to disjoint unit balls

Abstract

We prove Helly-type theorems for line transversals to disjoint unit balls in R-d. In particular, we show that a family of n >= 2d disjoint unit balls in R-d has a line transversal if, for some ordering < of the balls, any subfamily of 2d balls admits a line transversal consistent with <. We also prove that a family of n >= 4d-1 disjoint unit balls in R-d admits a line transversal if any subfamily of size 4d-1 admits a transversal.