A lower bound for the dispersion on the torus

We consider the volume of the largest axis-parallel box in the $d$-dimensional torus that contains no point of a given point set ${\cal P}_n$ with $n$ elements. We prove that, for all natural numbers $d,n$ and every point set ${\cal P}_n$, this volume is bounded from below by $min\{1,d/n\}$.This implies the same lower bound for the discrepancy on the torus.