Natan Rubin

We show that for any finite point set P in the plane and ϵ > 0 there exist O(1/ϵ3/2+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.