An Improved Bound for Weak Epsilon-Nets in the Plane

Abstract: We show that for any finite set $P$ of points in the plane and $\epsilon&gt;0$ there exist $$\displaystyle O\left(\frac{1}{\epsilon<sup>{3/2+\gamma}}\right)$$ points in ${\mathbb{R}}<sup>2$, for arbitrary small $\gamma&gt;0$, that pierce every convex set $K$ with $|K\cap P|\geq \epsilon |P|$. This is the first improvement of the bound of $$\displaystyle O\left(\frac{1}{\epsilon<sup>2}\right)$$ that was obtained in 1992 by Alon, B\'{a}r\'{a}ny, F\"{u}redi and Kleitman for general point sets in the plane.