Hadwiger and Helly-type theorems for disjoint unit spheres

Abstract

We prove Helly-type theorems for line transversals to disjoint unit balls in ${\mathbb{R}}^d$. In particular, we show that a family of $n\ge 2d$ disjoint unit balls in ${\mathbb{R}}^d$ has a line transversal if, for some ordering $\prec$ of the balls, any subfamily of 2d balls admits a line transversal consistent with $\prec$. We also prove that a family of $n\ge 4d-1$ disjoint unit balls in ${\mathbb{R}}^d$ admits a line transversal if any subfamily of size $4d-1$ admits a transversal.