A Lower Bound for Weak epsilon-Nets in High Dimension

A nite set N R d is a weak "-net for an n-point set X R d (with respect to convex sets) if it intersects each convex set K with jK \ Xj "n. It is shown that there are point sets X R d for which every weak 1 50-net has at least const e p d=2 points. Weak "-nets with respect to convex sets, as deened 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]. Alon et al. 1] proved that f (d; ") is nite for every d 1 and every " > 0. They established the bounds f (2; ") = O(" ?2) and f (d; ") C d " ?(d+1?(d)) , where C d depends only on d and (d) is a positive number tending to 0 (exponentially fast) as d ! 1. With a simpler proof, they obtained the slightly worse bound C 0 d " ?(d+1) , and here their proof yields C 0 d = d O(d). Chazelle et al. improved the bound for all xed dimensions d 3, to O(" ?d (log 1 ") b(d)) with a suitable constant b(d). It seems that no lower bound better than the obvious f (d; ") = (1 ") is known. Proving a lower bound superlinear in 1 " for some xed dimension remains a challenging open problem. In the present note, it is shown that for " xed and suuciently small and d ! 1, f (d; ") is at least e (p d). The best available upper bound in this situation is the d O(d) mentioned above, even for the particular set X used in the forthcoming proof of the lower bound. The lower bound is still meaningful up to " e ? p d , but for d xed it is entirely useless. For simplicity, we do the calculations for a particular value of ".