On the maximal number of edges of many faces in an arrangement (Q1071264)

An arrangement A of n lines partitions the real projective plane into regions, called faces. Let t(F) denote the number of lines each contributing an edge to the face F, \(a_ k(A)\) the maximal sum \(\sum^{k}_{j=1}t(Fj)\) which can be realized by k different faces \(F_ 1,F_ 2,...,F_ k\) in A, and \(a_ k(n)\) the maximum of \(a_ k(A)\) for all arrangements of n lines. The authors investigate asymptotic bounds for the maximal value of the sum \(\sum_{F\in {\mathcal L}}t({\mathcal L})\) for sets \({\mathcal L}\) of k faces in arrangements of n lines, and show that \(a_ k(n)=O(kn^{1/2}+n),\) \(a_ k(n)=O(nk^{1/2})\) and \(a_ k(n)=\Omega (n^{2/3}k^{2/3}+n).\) The paper contains a brief summary of earlier results and a discussion of related problems, and the authors point in the directions of further research.