  Gromov proved a cube inequality on the bound of distances between opposite
faces of a cube equipped with a positive scalar curvature metric in dimension
$\leq 8$ using minimal surface method. He conjectured that the cube inequality
also holds in dimension $\geq 9$. In this paper, we prove Gromov's cube
inequality in all dimensions with the optimal constant via Dirac operator
method. In fact, our proof yields a strengthened version of Gromov's cube
inequality, which does not seem to be accessible by minimal surface method.
