to specify three-dimensional polyhedra uniquely by the direction and perimeter of their facets. Minkowski's theorem and the uniqueness of this specification by direction and perimeter have a common generalization: whenever two three-dimensional convex polyhedra have the property that their facets have the same directions and no facet of one polyhedron can be translated into a proper subset of the facet with the same direction of the other polyhedron, the two polyhedra must be translates of each other. However, this version of the theorem does not generalize to higher dimensions. See also Alexandrov's uniqueness theorem Cauchy's theorem (geometry) References Polytopes Hermann Minkowski