An Improved Bound for Weak Epsilon-Nets in the Plane

08/08/2018

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by   Natan Rubin, et al.

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We show that for any finite set P of points in the plane and ϵ>0 there exist O(1/ϵ^3/2+γ) points in R^2, 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.