  Motivated by the search for reduced polytopes, we consider the following
question: For which polytopes exists a vertex-facet assignment, that is, a
matching between vertices and non-incident facets, so that the matching covers
either all vertices, or all facets? We provide general conditions for the
existence of such an assignment. We conclude that such exist for all simple and
simplicial polytopes, as well as all polytopes of dimension $d\le 6$. We
construct counterexample in all dimensions $d\ge 7$.
