On Gromov's dihedral extremality and rigidity conjectures

In this paper, we prove Gromov's dihedral extremality conjecture and dihedral rigidity conjecture regarding comparisons of scalar curvatures, mean curvatures and dihedral angles between two compact manifolds with corners possibly of different dimensions. As a consequence, we answer positively Gromov's dihedral extremality conjecture for polyhedra in all dimensions, and Gromov's dihedral rigidity conjecture for polyhedra in dimension three.