Empty Axis-Parallel Boxes

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