Emergent Mind
An Improved Bound for Weak Epsilon-Nets in the Plane
(1808.02686)
Published Aug 8, 2018
in
math.CO
,
cs.CG
,
and
cs.DM
Abstract
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 $\displaystyle O\left(\frac{1}{\epsilon2}\right)$ that was obtained in 1992 by Alon, B\'{a}r\'{a}ny, F\"{u}redi and Kleitman for general point sets in the plane.
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