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An improved bound for weak epsilon nets in the plane

Abstract:

We show that for any set $P$ of $n$ points in the plane and $\eps>0$ there exists a set of $O(1/\eps^{1.5+\gamma})$ points in the plane, for any \gamma>0, that pierce every convex set $K$ containing at least $\eps |P|$ points of P. This is the first improvement of the 1992 upper bound $O(1/eps^2)$ of Alon, Bárány, Füredi, and Kleitman.