The number of generalized balanced lines

TL;DR

This paper proves that for a specific configuration of red and blue points in the plane, there are at least as many balanced lines as red points, using novel rotation techniques.

Contribution

The paper introduces the concept of sliding rotations and proves a lower bound on the number of balanced lines in certain point sets.

Findings

At least r balanced lines exist for the given point configuration.

Introduction of sliding rotations as a new technique.

Generalization of rotation methods to prove combinatorial geometry results.

Abstract

Let $S$ be a set of $r$ red points and $b=r+2d$ blue points in general position in the plane, with $d\geq 0$. A line $\ell$ determined by them is said to be balanced if in each open half-plane bounded by $\ell$ the difference between the number of red points and blue points is $d$. We show that every set $S$ as above has at least $r$ balanced lines. The main techniques in the proof are rotations and a generalization, sliding rotations, introduced here.