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

Projections of polytopes on the plane and the generalized Baues problem