An Improved Bound for Weak Epsilon-nets in the Plane

We show that for any finite point set P in the plane and epsilon > 0 there exist O(1/epsilon(3/2+gamma)) points in R-2, for arbitrary small gamma > 0, that pierce every convex set K with vertical bar K boolean AND P vertical bar = epsilon vertical bar P vertical bar. This is the first improvement of the bound of O(1/epsilon 2) that was obtained in 1992 by Alon, Barany, Furedi, and Kleitman for general point sets in the plane.