Boundary Complexes of Convex Polytopes Cannot Be Characterized Locally

It is well known that there is no local criterion to decide the linear readability 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.