Near equipartitions of colored point sets

TL;DR

This paper proves the existence of disjoint convex sets or simplices containing points of all colors in colored point sets, extending geometric partitioning results to higher dimensions and multiple colors.

Contribution

It introduces new theorems guaranteeing partitions with points of all colors in both planar and higher-dimensional spaces.

Findings

Existence of n disjoint convex sets with mixed colors in 2D

Existence of n disjoint simplices with all colors in higher dimensions

Generalization of geometric partitioning theorems

Abstract

Suppose that $nk$ points in general position in the plane are colored red and blue, with at least $n$ points of each color. We show that then there exist $n$ pairwise disjoint convex sets, each of them containing $k$ of the points, and each of them containing points of both colors. We also show that if $P$ is a set of $n(d+1)$ points in general position in $\mathbb{R}^d$ colored by $d$ colors with at least $n$ points of each color, then there exist $n$ pairwise disjoint $d$-dimensional simplices with vertices in $P$, each of them containing a point of every color. These results can be viewed as a step towards a common generalization of several previously known geometric partitioning results regarding colored point sets.