# Bounding a global red-blue proportion using local conditions

## TL;DR

This paper establishes a tight lower bound on the global proportion of blue points relative to red points in the plane, based on local neighborhood conditions, and extends the result to higher dimensions and norms.

## Contribution

It introduces a novel local-to-global bound relating neighborhood conditions to overall point proportions, with optimality and generalizations to higher dimensions and Minkowski norms.

## Key findings

- Blue points are at least one fifth of red points under local conditions.
- The bound is proven to be optimal.
- Results extend to arbitrary dimensions and norms.

## Abstract

We study the following local-to-global phenomenon: Let $B$ and $R$ be two finite sets of (blue and red) points in the Euclidean plane $\mathbb{R}^2$. Suppose that in each "neighborhood" of a red point, the number of blue points is at least as large as the number of red points. We show that in this case the total number of blue points is at least one fifth of the total number of red points. We also show that this bound is optimal and we generalize the result to arbitrary dimension and arbitrary norm using results from Minkowski arrangements.