Stronger Bounds for Weak Epsilon-Nets in Higher Dimensions

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 \(d\geq 3\).