Piercing a set of disjoint balls by a line (Q5937250)

Given a family \({\mathcal F}_n\) of \(n\) disjoint \(d\)-dimensional balls in \({\mathbb R}^d\), with \(d\geq 2\), one denotes by \(\lambda=\lambda({\mathcal F}_n)\) the ratio of the maximum radius to the minimum radius among the balls in \({\mathcal F}_n\), and for any unit vector \(\vec{u}\) in \({\mathbb R}^d\) one denotes by \(f(\vec{u},{\mathcal F}_n)\) the maximum number of balls in \({\mathcal F}_n\) a line parallel to \(\vec{u}\) can intersect. NEWLINENEWLINENEWLINEImproving results from \textit{N. Alon, M. Katchalski} and \textit{W. R. Pulleyblank} [Discrete Comput. Geom. 4, 239-243 (1989; Zbl 0719.52006)] and from their own paper [J. Comb. Theory, Ser. A 90, 235-240 (2000; Zbl 0956.52004)], the authors show that, on the one hand for any family \({\mathcal F}_n\) there is a unit vector \(\vec{u}\), such that \(f(\vec{u}, {\mathcal F}_n)\) is at most \(O(\sqrt{(1+\log\lambda)n\log n})\), and on the other hand, that for any \(n>d\), there is a family \({\mathcal F}_n\) such that \(f(\vec{u},{\mathcal F}_n)\geq n-d+1\) for every unit vector \(\vec{u}\).