Empty Axis-Parallel Boxes

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 -> infinity and d is fixed. In the opposite direction, we give a construction without an empty axis-parallel box of volume O(d(2) logd/n). These improve on the previous best bounds of Omega(logd/n) and O(2(7d)/n), respectively.