Balanced convex partitions of lines in the plane

TL;DR

This paper extends a ham sandwich theorem for lines in the plane, demonstrating optimal bounds for convex partitions that evenly distribute line intersections from two line sets.

Contribution

It introduces a new convex partitioning method for line sets in the plane, extending the ham sandwich theorem with optimal bounds on line distribution.

Findings

Partition into r convex regions with balanced line intersections

Optimal dependence on the number of lines n

Extension of a known geometric theorem

Abstract

We prove an extension of a ham sandwich theorem for families of lines in the plane by Dujmovi\'{c} 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 $r$ convex regions such 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.