  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 the smoothing construction,
thereby providing a detailed proof of Gromov's theorem. The proof of the
rigidity statement also uses a deep estimate due to Fefferman and Phong.
