A Lower Bound for Weak Epsilon-Nets in High Dimension

A finite set N ae R d is a weak "-net for an n-point set X ae R d (with respect to convex sets) if it intersects each convex set K with jK " X j "n. It is shown that there are point sets X ae R d for which every weak 1 50 -net has at least const \Delta e p d=2 points. Weak "-nets with respect to convex sets, as defined in the abstract, were introduced by Haussler and Welzl [6] and later applied in results in discrete geometry, most notably in the spectacular proof of the Hadwiger--Debrunner (p; q)-conjecture by Alon and Kleitman [2]. For a finite X ae R d , let f(X; ") denote the smallest size of a weak "-net for X, 0 ! " ! 1, and let f(d; ") = supff(X; ") : X ae R d finiteg: Alon et al. [1] proved that f(d; ") is finite for every d 1 and every " ? 0. They established the bounds f(2; ") = O(" \Gamma2 ) and f(d; ") C d " \Gamma(d+1\Gammaffi(d)) , where C d depends only on d and ffi(d) is a positive number tending to 0 (exponentially fast) as d !1. With a simpler..