A proof of Gromov's cube inequality on scalar curvature

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 a 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 a Dirac operator method.