Piercing a Set of Disjoint Balls by a Line

Let Fn denote a family of disjoint n balls in Rd(d⩾2), and let λ=λ(Fn) denote the ratio (maximum radius)/(minimum radius) among the balls in Fn. We prove that (1) there is a unit vector \vec{u} such that every line parallel to \vec{u} intersects at most O((1+log λ) n log n) balls of Fn, and (2) there is a family Fn such that for any unit vector \vec{u} there is a line parallel to \vec{u} that intersects at least n−d+1 balls of Fn.