On Gromov's rigidity theorem for polytopes with acute angles

In his ``Four Lectures", Gromov conjectured a scalar curvature extremality property of convex polytopes. Moreover, Gromov outlined a proof of the conjecture in the special case when the dihedral angles are acute. Gromov's argument relies on Dirac operator techniques together with a smoothing construction. In this paper, we give the details of such a smoothing construction, thereby providing a detailed proof of Gromov's theorem.