A Lower Bound for Weak ɛ -Nets in High Dimension

A finite set N ⊂ Rd is a weak ε-net for an n -point set X ⊂ Rd (with respect to convex sets) if it intersects each convex set K with |K ∩ X| ≥ ε n . It is shown that there are point sets X ⊂ Rd for which every weak ε -net has at least const ⋅\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $e^{\sqrt{d/2}}$ \end{document} points. This distinguishes the behavior of weak ε -nets with respect to convex sets from ε -nets with respect to classes of shapes like balls or ellipsoids in Rd, where the size can be bounded from above by a polynomial function of d and ε.