On the size of the largest empty box amidst a point set

Abstract

The problem of finding the largest empty axis-parallel box amidst a point configuration is a classical problem in computational geometry. It is known that the volume of the largest empty box is of asymptotic order 1∕n for n→∞ and fixed dimension d. However, it is natural to assume that the volume of the largest empty box increases as d gets larger. In the present paper we prove that this actually is the case: for every set of n points in [0,1]d there exists an empty box of volume at least cdn−1, where cd→∞ as d→∞. More precisely, cd is at least of order roughly logd.