平面弱ε-网的一个改进界
An Improved Bound for Weak Epsilon-Nets in the Plane
摘要：
证明了对平面上任意有限集$P$和$ε>0$，都存在${\mathbb{R}}^2$中的$\displaystyle O\left(\frac{1}{ε^{3/2+γ}}\right)$点，对任意小$γ>0$，都存在用$|K\cap P|\geq ε|P|$穿透每个凸集$K$的$\displaystyle O\left(\frac{1}{ε^{3/2+γ}}\right)$点.这是$\displaystyle O\left(\frac{1}{ε^2}\right)$在1992年由Alon，Bárány，Füredi和Kleitman对平面上的一般点集得到的界的第一个改进。
英文摘要：
We show that for any finite set $P$ of points in the plane and $ε>0$ there exist $\displaystyle O\left(\frac{1}{ε^{3/2+γ}}\right)$ points in ${\mathbb{R}}^2$, for arbitrary small $γ>0$, that pierce every convex set $K$ with $|K\cap P|\geq ε|P|$. This is the first improvement of the bound of $\displaystyle O\left(\frac{1}{ε^2}\right)$ that was obtained in 1992 by Alon, Bárány, Füredi and Kleitman for general point sets in the plane.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
