Perfect vector sets, properly overlapping partitions, and largest empty box

We revisit the following problem (along with its higher dimensional variant): Given a set S of n points inside an axis-parallel rectangle U in the plane, find a maximum-area axis-parallel sub-rectangle that is contained in U but contains no points of S. (I) We present an algorithm that finds a large empty box amidst n points in [0, 1]: a box whose volume is at least log d 4(n+log d) can be computed in O(n+ d log d) time. (II) To better analyze the above approach, we introduce the concepts of perfect vector sets and properly overlapping partitions, in connection to the minimum volume of a maximum empty box amidst n points in the unit hypercube [0, 1], and derive bounds on their sizes.