A theorem about antiprisms

Let P be a polytope in Rⁿ containing the origin in its interior, and let P* be the algebraic dual polytope of P. Let Q Rⁿ x [0,1] be the (n+1)-dimensional polytope that is the convex hull of P x {1} and P* x {0}. For each face F of P, let Q(F) denote the convex hull of F x {1} and F* x {0}, where F* is the dual face of P*. Then Q is an antiprism if the set of facets of Q is precisely the collection {Q (F)} for all faces F of P. If Q is an antiprism, the correspondence between primal and dual faces of P and P* is manifested in the facets of Q. In this paper, necessary and sufficient conditions for the existence of antiprisms are stated and proved.