# Dense point sets with many halving lines

## TL;DR

This paper constructs dense point sets with exponentially many halving lines in the plane and hyperplanes in higher dimensions, surpassing previous bounds and matching known lower bounds asymptotically.

## Contribution

It introduces a novel construction of dense point sets with significantly more halving lines than previously known, applicable in any dimension.

## Key findings

- Constructed dense point sets with $ne^{ ilde{ ext{O}}(
oot{ ext{log n}})$ halving lines in 2D.
- Extended the construction to higher dimensions with $n^{d-1}e^{ ilde{ ext{O}}(
oot{ ext{log n}})}$ halving hyperplanes.
- Achieved bounds asymptotically matching the best known lower bounds for general point sets.

## Abstract

A planar point set of $n$ points is called {\em $\gamma$-dense} if the ratio of the largest and smallest distances among the points is at most $\gamma\sqrt{n}$. We construct a dense set of $n$ points in the plane with $ne^{\Omega\left({\sqrt{\log n}}\right)}$ halving lines. This improves the bound $\Omega(n\log n)$ of Edelsbrunner, Valtr and Welzl from 1997. Our construction can be generalized to higher dimensions, for any $d$ we construct a dense point set of $n$ points in $\mathbb{R}^d$ with $n^{d-1}e^{\Omega\left({\sqrt{\log n}}\right)}$ halving hyperplanes. Our lower bounds are asymptotically the same as the best known lower bounds for general point sets.