An elementary proof of the Minkowski and some other theorems on convex polyhedra

A convex polyhedron is uniquely determined, up to translations, by the directions and the areas of its faces. By the direction o f a face, we mean the direction of the outward normal to the face.

The proof of this theorem, as given by Minkowski, is based on the Brunn inequality which needs by far nonelementary and intricate reasoning for its demonstration. This fact, noted by Minkowski in the excerpt quoted above, distinguishes his theorem from the other results of the theory of polyhedra.