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 $|K\cap P|\geq \epsilon |P|$. Let $d\geq 3$. We show that for any finite point set in ${\mathbb R}^d$, and any $\epsilon>0$, there exist a weak $\epsilon$-net of cardinality $\displaystyle O\left(\frac{1}{\epsilon^{d-1/2+\gamma}}\right)$, where $\gamma>0$ is an arbitrary small constant. This is the first improvement of the bound of $\displaystyle O^*\left(\frac{1}{\epsilon^d}\right)$ that was obtained in 1994 by Chazelle, Edelsbrunner, Grigni, Guibas, Sharir, and Welzl for general point sets in dimension $d\geq 3$.

中文翻译：

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较大尺寸的弱Epsilon网的边界更强

给定$ {\ mathbb R} ^ d $中的有限点集$ P $，并且$ \ epsilon> 0 $，我们说$ N \ subseteq {\ mathbb R} ^ d $是一个弱的$ \ epsilon $ -net如果它用$ | K \ cap P | \ geq \ epsilon | P | $刺穿每个凸集$ K $。让$ d \ geq 3 $。我们表明，对于$ {\ mathbb R} ^ d $中设置的任何有限点，以及$ \ epsilon> 0 $，存在弱的$ \ epsilon $-基数网$ \ displaystyle O \ left（\ frac { 1} {\ epsilon ^ {d-1 / 2 + \ gamma}} \ right）$，其中$ \ gamma> 0 $是一个任意小的常数。这是对$ \ displaystyle O ^ * \ left（\ frac {1} {\ epsilon ^ d} \ right）$边界的首次改进，该边界是在1994年由Chazelle，Edelsbrunner，Grigni，Guibas，Sharir和对于维数为$ d \ geq 3 $的一般点集的Welzl。