  Given a finite point set $P$ in ${\mathbb R}^d$, and $\epsilon>0$ we say that
$N\subseteq{ \mathbb R}^d$ is a weak $\epsilon$-net if it pierces every convex
set $K$ with $|K\cap P|\geq \epsilon |P|$. We show that for any finite point
set in dimension $d\geq 3$, and any $\epsilon>0$, one can construct a weak
$\epsilon$-net whose cardinality is $\displaystyle
O^*\left(\frac{1}{\epsilon^{2.558}}\right)$ in dimension $d=3$, and
$\displaystyle o\left(\frac{1}{\epsilon^{d-1/2}}\right)$ in all dimensions
$d\geq 4$.
  To be precise, our weak $\epsilon$-net has cardinality $\displaystyle
O\left(\frac{1}{\epsilon^{\alpha_d+\gamma}}\right)$ for any $\gamma>0$, with
  $$
  \alpha_d=
  \left\{
  \begin{array}{l}
  2.558 & \text{if} \ d=3
  \\3.48 & \text{if} \ d=4
  \\\left(d+\sqrt{d^2-2d}\right)/2 & \text{if} \ d\geq 5.
  \end{array}\right\}
  $$
  This is the first significant improvement of the bound of $\displaystyle
\tilde{O}\left(\frac{1}{\epsilon^d}\right)$ that was obtained in 1993 by
Chazelle, Edelsbrunner, Grigni, Guibas, Sharir, and Welzl for general point
sets in dimension $d\geq 3$.
