The author shows under certain conditions that the minimal surface, decomposing a given polyhedron into parts of given volumes, is not polyhedral and lists some characteristic local properties of the optimal polyhedral decomposition, assuming such decomposition exists, of a given polyhedron with a given volume. A cell system defined by the shortest net is called optimal decomposition. He gives conditions that hold for the inner faces of an optimal dividing surface and presents some corollaries derived from the conditions concerning the structure of an optimal polyhedral decomposition of a polyhedron in \(E^3\). In the first two of these corollaries it is stated that the optimal polyhedral decomposition of a polyhedron into parts of given volumes contains respectively no inner cell and no face with merely inner vertices. In the third one it is stated that when the optimal solution of a slab decomposition problem is polyhedral then it has no inner nodes. In corollaries 4 and 5 the polyhedron that will be decomposed into parts of given volumes is considered to be convex, and in an optimal polyhedral decomposition it is respectively expressed that an inner face has at most two inner vertices and an inner node of the dividing surface has at most one inner neighbor. Finally a detailed classification is given when the polyhedron to be decomposed is taken convex.