An Improved Bound for Weak Epsilon-nets in the Plane

Abstract

We show that for any finite point set P in the plane and ϵ > 0 there exist \( O(\tfrac{1}{{\epsilon }^{3/2+\gamma }}) \) points in ℝ 2 , for arbitrary small γ > 0, that pierce every convex set K with | K ∩ P |> ϵ | P |. This is the first improvement of the bound of \( O(\tfrac{1}{{\epsilon }^2}) \) that was obtained in 1992 by Alon, Bárány, Füredi, and Kleitman for general point sets in the plane.