加权ε-网
Weighted Epsilon-Nets
摘要：
受Bukh和Nivasch最近关于单侧$\varepsilon$ -逼近的工作的启发，我们引入了\emph{weighted $\varepsilon$-nets}的概念。它是$\mathbb{R}^d$中点集近似的一个几何概念，类似于$\varepsilon$ -网和$\varepsilon$ -近似，比前者强，比后者弱。主要思想是小集合可以包含许多点，而大集合必须包含加权$\varepsilon$ -网的许多点。本文分析了弱加权$\varepsilon$ -网关于凸集和轴平行盒的性质，并给出了加权$\varepsilon$ -网关于2和3的$\varepsilon$的上下界.其中一些界限也适用于经典的$\varepsilon$ -网。
英文摘要：
Motivated by recent work of Bukh and Nivasch on one-sided $\varepsilon$-approximants, we introduce the notion of \emph{weighted $\varepsilon$-nets}. It is a geometric notion of approximation for point sets in $\mathbb{R}^d$ similar to $\varepsilon$-nets and $\varepsilon$-approximations, where it is stronger than the former and weaker than the latter. The main idea is that small sets can contain many points, whereas large sets must contain many points of the weighted $\varepsilon$-net. In this paper, we analyze weak weighted $\varepsilon$-nets with respect to convex sets and axis-parallel boxes and give upper and lower bounds on $\varepsilon$ for weighted $\varepsilon$-nets of size two and three. Some of these bounds apply to classical $\varepsilon$-nets as well.
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。
