  In this paper, we develop a new index theory for manifolds with polyhedral
boundary. As an application, we prove Gromov's dihedral extremality conjecture
regarding comparisons of scalar curvatures, mean curvatures and dihedral angles
between two compact manifolds with polyhedral boundary in all dimensions. We
also prove Gromov's dihedral rigidity conjecture for a class of positively
curved manifolds with polyhedral boundary in all dimensions.
