Boundary Complexes of Convex Polytopes cannot Be Characterized Locally
Bernd Sturmfels · Journal of the London Mathematical Society · 1987
It is well known that there is no local criterion to decide the linear realizability of matroids or oriented matroids. We use the set-up of chirotopes or oriented matroids to derive a similar result in the context of convex polytopes. There is no local criterion to decide whether a combinatorial sphere is polytopal. The proof is based on a construction technique for rigid chirotopes. These correspond, in the realizable case, to convex polytopes whose internal combinatorial structure is completely determined by its face lattice. So, a rigid chirotope is realizable over a field F if and only if its face-lattice is F-polytopal. Furthermore we prove that for every proper subfield F of the field A of real algebraic numbers there exists a 6-polytope which is not realizable over F.