平面中直线的ε-网的最优下界
Optimal lower bounds for epsilon-nets for lines in the plane
中文总结 AI 辅助
该研究证明平面直线的ε-网存在最优下界，其基数为Ω(1/ε · log(1/ε))，匹配经典上界并证实Alon的预测。
中文摘要 AI 辅助
我们证明，对于任意小的正ε，存在有限平面点集P，使得由直线在P上诱导的范围空间的每个ε-网都具有Ω(1/ε · log(1/ε))的基数。这与Haussler和Welzl提出的有界VC-维数范围空间的经典上界匹配，并证实了Alon的一个预测。
英文摘要
We prove that, for arbitrarily small positive $\varepsilon$, there is a finite planar point set $P$ such that every $\varepsilon$-net for the range space induced on $P$ by straight lines has cardinality $Ω\bigl(1/\varepsilon \cdot \log(1/\varepsilon)\bigr)$. This matches the classical upper bound for range spaces with bounded VC-dimension due to Haussler and Welzl and confirms a prediction of Alon.
