An Improved Bound for Weak Epsilon-Nets in the Plane

TL;DR

This paper improves the upper bound on the size of weak epsilon-nets in the plane, reducing it from quadratic to nearly cubic root dependence on 1/epsilon, for arbitrary small gamma.

Contribution

It presents the first improvement over the long-standing quadratic bound for weak epsilon-nets in the plane since 1992.

Findings

New bound of O(1/epsilon^{3/2+γ}) points for weak epsilon-nets

First improvement over the 1992 quadratic bound

Applicable for any small γ > 0

Abstract

We show that for any finite set $P$ of points in the plane and $\epsilon>0$ there exist $\displaystyle O\left(\frac{1}{\epsilon^{3/2+\gamma}}\right)$ points in ${\mathbb{R}}^2$, for arbitrary small $\gamma>0$, that pierce every convex set $K$ with $|K\cap P|\geq \epsilon |P|$. This is the first improvement of the bound of $\displaystyle O\left(\frac{1}{\epsilon^2}\right)$ that was obtained in 1992 by Alon, B\'{a}r\'{a}ny, F\"{u}redi and Kleitman for general point sets in the plane.