  We prove the following comparison theorem for metrics with nonnegative scalar
curvature, also known as the dihedral rigidity conjecture by Gromov: for $n\le
7$, if an $n$-dimensional prism has nonnegative scalar curvature and weakly
mean convex faces, then its dihedral angle cannot be everywhere not larger than
its Euclidean model, unless it is isometric to an Euclidean prism. The proof
relies on constructing certain free boundary minimal hypersurface in a
Riemannian polyhedron, and extending a dimension descent idea of Schoen-Yau.
Our result is a localization of the positive mass theorem.
