Given a finite point set $P$ in ${\mathbb R}^d$, and $\epsilon>0$ we say that $N\subseteq{ \mathbb R}^d$ is a weak $\epsilon$-net if it pierces every convex set $K$ with $|K\cap P|\geq \epsilon |P|$. We show that for any finite point set in dimension $d\geq 3$, and any $\epsilon>0$, one can construct a weak $\epsilon$-net whose cardinality is $\displaystyle O^*\left(\frac{1}{\epsilon^{2.558}}\right)$ in dimension $d=3$, and $\displaystyle o\left(\frac{1}{\epsilon^{d-1/2}}\right)$ in all dimensions $d\geq 4$. To be precise, our weak $\epsilon$-net has cardinality $\displaystyle O\left(\frac{1}{\epsilon^{\alpha_d+\gamma}}\right)$ for any $\gamma>0$, with $$ \alpha_d= \left\{ \begin{array}{l} 2.558 & \text{if} \ d=3 \\3.48 & \text{if} \ d=4 \\\left(d+\sqrt{d^2-2d}\right)/2 & \text{if} \ d\geq 5. \end{array}\right\} $$ This is the first significant improvement of the bound of $\displaystyle \tilde{O}\left(\frac{1}{\epsilon^d}\right)$ that was obtained in 1993 by Chazelle, Edelsbrunner, Grigni, Guibas, Sharir, and Welzl for general point sets in dimension $d\geq 3$.

翻译：给定有限点集 $P \subseteq {\mathbb R}^d$ 和 $\epsilon>0$，若 $N\subseteq{\mathbb R}^d$ 与每个满足 $|K\cap P|\geq \epsilon|P|$ 的凸集 $K$ 均相交，则称 $N$ 为一个弱ε-网。我们证明：对任意维度 $d\geq 3$ 的有限点集及任意 $\epsilon>0$，可构造一个弱ε-网，其基数在 $d=3$ 时达到 $\displaystyle O^*\left(\frac{1}{\epsilon^{2.558}}\right)$，在 $d\geq 4$ 时达到 $\displaystyle o\left(\frac{1}{\epsilon^{d-1/2}}\right)$。更精确地，对任意 $\gamma>0$，该弱ε-网的基数为 $\displaystyle O\left(\frac{1}{\epsilon^{\alpha_d+\gamma}}\right)$，其中 $$ \alpha_d= \left\{ \begin{array}{l} 2.558 & \text{若} \ d=3 \\3.48 & \text{若} \ d=4 \\\left(d+\sqrt{d^2-2d}\right)/2 & \text{若} \ d\geq 5. \end{array}\right\} $$ 这是对1993年Chazelle、Edelsbrunner、Grigni、Guibas、Sharir和Welzl针对维度 $d\geq 3$ 的一般点集所获的界 $\displaystyle \tilde{O}\left(\frac{1}{\epsilon^d}\right)$ 的首次重大改进。