A Helly-Type Theorem for Line Transversals to Disjoint Unit Balls

Abstract

Abstract. Let F be a family of disjoint unit balls in R 3 . We prove that there is a Helly-number n 0 ≤ 46 , such that if every n 0 members of F ( | F | ≥ n 0 ) have a line transversal, then F has a line transversal. In order to prove this we prove that if the members of F can be ordered in a way such that every 12 members of F are met by a line consistent with the ordering, then F has a line transversal. The proof also uses the recent result on geometric permutations for disjoint unit balls by Katchalski, Suri, and Zhou.