An Improved Bound for Weak Epsilon-Nets in the Plane

We show that for any finite point set P in the plane and ε>0 there exist O(1/ε
^3/2+γ
) points, 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(1)ε
²
) that was obtained in 1992 by Alon, Bárány, Füredi and Kleitman for general point sets in the plane.