For a point set of $n$ elements in the $d$-dimensional unit cube and a class of test sets we are interested in the largest volume of a test set which does not contain any point. For all natural numbers $n$, $d$ and under the assumption of a $delta$-cover with cardinality $\vert \Gamma_\delta \vert$ we prove that there is a point set, such that the largest volume of such a test set without any point is bounded by $\frac{ \log \vert \Gamma_\delta \vert}{ n} + \delta$. For axis-parallel boxes on the unit cube this leads to a volume of at most $\frac{ 4d}{ n}\log(\frac{ 9n}{ d})$ and on the torus to $\frac{ 4d}{ n}\log (2n)$.
Categories: cs.CG math.NA