Stronger bounds for weak epsilon-nets in higher dimensions

Abstract

Given a finite point set P in ℝd, and >0 we say that N⊆ ℝd is a weak -net if it pierces every convex set K with |K∩ P|≥ є |P|. Let d≥ 3. We show that for any finite point set in ℝd, and any є>0, there exist a weak -net of cardinality O(1/єd−1/2+γ), where γ>0 is an arbitrary small constant. This is the first improvement of the bound of O*(1/єd) that was obtained in 1993 by Chazelle, Edelsbrunner, Grigni, Guibas, Sharir, and Welzl for general point sets in dimension d≥ 3.