Abstract

A line ℓ is a transversal to a family F of convex objects in R d if it intersects every member of F. If, in addition, ℓ is an isolated point of the space of line transversals to F, we say that it is pinned by F. It is known that if a line is pinned by at least 2d pairwise disjoint balls in R d then one can always remove one of the balls and keep the line pinned. We show that the constant 2d is tight for all dimensions, that is, there exist families of 2d − 1 disjoint balls in R d that pin a line such that if any ball is removed, the family no longer pins the line. As a consequence, we get that the Helly number of sets of line transversals to disjoint unit balls in R d must increase with the dimension, which answers a question of Danzer from 1957.

Citations