  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.
