Balanced Convex Partitions of Lines in the Plane

Abstract

We prove an extension of a ham sandwich theorem for families of lines in the plane by Dujmović and Langerman. Given two sets A, B of n lines each in the plane, we prove that it is possible to partition the plane into r closed convex regions so that the following holds. For each region C of the partition there is a subset of \(c_r n^{1/r}\) lines of A whose pairwise intersections are in C, and the same holds for B. In this statement \(c_r\) only depends on r. We also prove that the dependence on n is optimal. For a single set A of n lines, we prove that there exists a partition of the plane into r parts using \(r-1\) vertical lines such that each region contains the pairwise intersections of a set of \(n^{1/r}\) lines of A. The value \(n^{1/r}\) is optimal.