Expected dispersion of uniformly distributed points

The dispersion of a point set in [0,1]d is the volume of the largest axis parallel box inside the unit cube that does not intersect the point set. We study the expected dispersion with respect to a random set of n points determined by an i.i.d. sequence of uniformly distributed random variables. Depending on the number of points n and the dimension d we provide an upper and a lower bound of the expected dispersion. In particular, we show that the minimal number of points required to achieve an expected dispersion less than ε∈(0,1) depends linearly on the dimension d.