On the structure of faces of three-dimensional polytopes

摘要

In Euclidean three-space, we consider two-dimensional polyhedra that are homeomorphic to closed surfaces. The structure of an arbitrary face of such a polyhedron is studied in detail. In particular, we prove the following main theorem. If a two-dimensional polyhedron lies in Euclidean three-space and is isometric to the surface of a convex three-dimensional polytope, then all the faces of the polyhedron are convex polygons.