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 ${\mathcal{P}}_n$ with $n$ elements. We prove that, for all natural numbers $d,n$ and every point set ${\mathcal{P}}_n$, this volume is bounded from below by $\min \{1,d\mathrm{/}n\}$. This implies the same lower bound for the discrepancy on the torus.