Projections of polytopes on the plane and the generalized Baues problem

Given an affine projection pi : P --> Q of a d-polytope P onto a polygon Q, it is proved that the poset of proper polytopal subdivisions of Q which ar e induced by pi has the homotopy type of a sphere of dimension d - 3 if pi maps all vertices of P into the boundary of Q. This result, originally conj ectured by Reiner, is an analogue of a result of Billera, Kapranov and Stur mfels on cellular strings on polytopes and explains the significance of the interior point of Q present in the counterexample to their generalized Bau es conjecture, constructed by Rambau and Ziegler.