# On circles enclosing many points

## TL;DR

This paper investigates geometric properties of point sets in the plane, establishing bounds on points enclosed by circles through pairs of points and analyzing higher order Voronoi diagrams.

## Contribution

It provides new bounds on circle enclosures in two-colored point sets and explores properties of higher order Voronoi diagrams, extending previous work.

## Key findings

- Existence of red-blue point pairs with circles enclosing many points
- Existence of point pairs with circles enclosing few points
- Analysis of collinear edges in higher order Voronoi diagrams

## Abstract

We prove that every set of $n$ red and $n$ blue points in the plane contains a red and a blue point such that every circle through them encloses at least $n(1-\frac{1}{\sqrt{2}}) -o(n)$ points of the set. This is a two-colored version of a problem posed by Neumann-Lara and Urrutia. We also show that every set $S$ of $n$ points contains two points such that every circle passing through them encloses at most $\lfloor{\frac{2n-3}{3}}\rfloor$ points of $S$. The proofs make use of properties of higher order Voronoi diagrams, in the spirit of the work of Edelsbrunner, Hasan, Seidel and Shen on this topic. Closely related, we also study the number of collinear edges in higher order Voronoi diagrams and present several constructions.