Fiber Polytopes for the Projections between Cyclic Polytopes

The cyclic polytope C(n, d) is the convex hull of any n points on the moment curve {(t, t2, . . . , td ) : t ∈ R} in Rd . For d ′ > d , we consider the fiber polytope (in the sense of Billera and Sturmfels [6]) associated to the natural projection of cyclic polytopes π : C(n, d ) → C(n, d) which ‘forgets’ the last d ′ − d coordinates. It is known that this fiber polytope has face lattice indexed by the coherent polytopal subdivisions of C(n, d) which are induced by the map π . Our main result characterizes the triples (n, d, d ) for which the fiber polytope is canonical in either of the following two senses: