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slides
An Improved Bound for Weak Epsilon-Nets in the Plane
Authors
Natan Rubin
Publication date
8 August 2018
Publisher
View
on
arXiv
Abstract
We show that for any finite set
P
P
P
of points in the plane and
ϵ
>
0
\epsilon>0
ϵ
>
0
there exist
O
(
1
ϵ
3
/
2
+
γ
)
\displaystyle O\left(\frac{1}{\epsilon^{3/2+\gamma}}\right)
O
(
ϵ
3/2
+
γ
1
​
)
points in
R
2
{\mathbb{R}}^2
R
2
, for arbitrary small
γ
>
0
\gamma>0
γ
>
0
, that pierce every convex set
K
K
K
with
∣
K
∩
P
∣
≥
ϵ
∣
P
∣
|K\cap P|\geq \epsilon |P|
∣
K
∩
P
∣
≥
ϵ
∣
P
∣
. This is the first improvement of the bound of
O
(
1
ϵ
2
)
\displaystyle O\left(\frac{1}{\epsilon^2}\right)
O
(
ϵ
2
1
​
)
that was obtained in 1992 by Alon, B\'{a}r\'{a}ny, F\"{u}redi and Kleitman for general point sets in the plane.Comment: A preliminary version to appear in the proceedings of FOCS 201
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oai:arXiv.org:1808.02686
Last time updated on 08/09/2018