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Weighted Epsilon-Nets

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.