Preassigning the shape of projections of convex polytopes

The shape of any facet of a 3-polytope can be preassigned. That is, given any 3-polytope P a facet F of P and a polygon F’ with the same number of sides as F, there is an isomorphic 3-polytope P’ such that F is a facet of P’ and the isomorphism takes F onto F (see [ 21). It is also known that given any simple circuit C in the graph of a 3-polytope P, there is an isomorphic 3-polytope P’ such that an orthogonal projection of P’ will take the corresponding circuit onto the boundary of the projection ([ 1)). Griinbaum has asked if given a circuit C of length n in a 3-polytope P and an n-gon A, there is an isomorphic 3-polytope P’ for which the corresponding circuit projects orthogonally onto A [4]. We shall show that there is not always such a 3-polytope P’.