Polyhedral spheres

In the preceding chapter, we dealt with boundary complexes of convex polytopes. They consist of cell decompositions of topological spheres, the cells again being convex polytopes. If, however, any cell decomposition of a topological sphere is given, there need not exist a convex polytope with isomorphic (in the sense of inclusion of cells) boundary complex. We shall present counter-examples in section 4 below. In fact, one of the major unsolved problems in convex polytope theory is to find necessary and sufficient conditions for a cell-composed sphere to be isomorphic to the boundary complex of a polytope (Steinitz problem).