Balanced lines in two-coloured point sets

TL;DR

This paper proves that in a two-colored point set with an even total, there are at least as many balanced lines as the smaller color class, refining previous results for equal-sized sets.

Contribution

The paper extends prior work by establishing a lower bound on the number of balanced lines for arbitrary blue and red point set sizes.

Findings

At least min{|B|, |R|} balanced lines exist in the set.

The result generalizes previous findings for equal-sized point sets.

Provides a new bound that applies to all two-colored point sets in general position.

Abstract

Let $B$ and $R$ be point sets (of {\em blue} and {\em red} points, respectively) in the plane, such that $P:=B\cup R$ is in general position, and $|P|$ is even. A line $\ell$ is {\em balanced} if it spans one blue and one red point, and on each open halfplane of $\ell$, the number of blue points minus the number of red points is the same. We prove that $P$ has at least $\min \{|B|,|R|\} $ balanced lines. This refines a result by Pach and Pinchasi, who proved this for the case $|B|=|R|$.