Stronger Bounds for Weak Epsilon-Nets in Higher Dimensions

04/26/2021

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by   Natan Rubin, et al.

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Given a finite point set P in ℝ^d, and ϵ>0 we say that N⊆ℝ^d is a weak ϵ-net if it pierces every convex set K with |K∩ P|≥ϵ |P|. Let d≥ 3. We show that for any finite point set in ℝ^d, and any ϵ>0, there exist a weak ϵ-net of cardinality O(1/ϵ^d-1/2+γ), where γ>0 is an arbitrary small constant. This is the first improvement of the bound of O^*(1/ϵ^d) that was obtained in 1994 by Chazelle, Edelsbrunner, Grigni, Guibas, Sharir, and Welzl for general point sets in dimension d≥ 3.