Stronger Bounds for Weak Epsilon-Nets in Higher Dimensions

Given a finite point set $P$ in $R^d$, and $\eps>0$ we say that a point set $N$

in  $R^d$ is a weak $\eps$-net if it pierces every convex set $K$ with 

$|K \cap P| \geq \eps |P|$.

Let $d\geq 3$. We show that for any finite point set in $R^d$, and any

$\eps>0$, there exists a weak $\eps$-net of cardinality $o(1/\eps^{d-1/2})$,

where \delta>0 is an arbitrary small constant. 

This is the first improvement of the bound of $O^*(1/\eps^d)$ that was obtained

in 1993 by Chazelle, Edelsbrunner, Grigni, Guibas, Sharir, and Welzl for

general point sets in dimension $d \geq 3$.