  We show that, for every set of $n$ points in the $d$-dimensional unit cube,
there is an empty axis-parallel box of volume at least $\Omega(d/n)$ as
$n\to\infty$ and $d$ is fixed. In the opposite direction, we give a
construction without an empty axis-parallel box of volume $O(d^2\log d/n)$.
These improve on the previous best bounds of $\Omega(\log d/n)$ and
$O(2^{7d}/n)$ respectively.
