Stronger bounds for weak epsilon-nets in higher dimensions

Abstract: Given a finite point set $P$ in $R^d$, and $\varepsilon>0$ we say that a point set $N$ in $R^d$ is a weak $\varepsilon$-net if it pierces every convex set $K$ with $|K\cap P|\geq \varepsilon |P|$. Let $d\geq 3$. We show that for any finite point set in $R^d$, and any $\varepsilon>0$, there exists a weak $\varepsilon$-net of cardinality $O(1/\varepsilon^{d-1/2+\delta})$, where $\delta>0$ is an arbitrary small constant.

This is the first improvement of the bound of $O^*(1/\varepsilon^d)$ that was obtained in 1993 by Chazelle, Edelsbrunner, Grigni, Guibas, Sharir, and Welzl for general point sets in dimension $d\geq 3$.

computational geometrydiscrete mathematicscommutative algebracombinatorics

Audience: researchers in the topic

Copenhagen-Jerusalem Combinatorics Seminar

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