Bounds for the Average $L^p$-Extreme and the $L^\infty$-Extreme Discrepancy

Abstract

The extreme or unanchored discrepancy is the geometric discrepancy of point sets in the $d$-dimensional unit cube with respect to the set system of axis-parallel boxes. For $2\leq p < \infty$ we provide upper bounds for the average $L^p$-extreme discrepancy. With these bounds we are able to derive upper bounds for the inverse of the $L^\infty$-extreme discrepancy with optimal dependence on the dimension $d$ and explicitly given constants.