Cyclic Polytopes and Oriented Matroids

Consider the moment curve in the real euclidean space Rddefined parametrically by the map γ: R → Rd,t ∣ → γ(t) = (t, t2,⋯ , td). The cyclic d -polytopeCd (t1,⋯ , tn) is the convex hull ofn & d different points on this curve. The matroidal analogs are the alternating oriented uniform matroids. A polytope (resp. matroid polytope) is called cyclic if its face lattice is isomorphic to that ofCd (t1,⋯ , tn). We give combinatorial and geometrical characterizations of cyclic (matroid) polytopes. A simple evenness criterion determining the facets ofCd (t1,⋯ , tn) was given by Gale . We characterize the admissible orderings of the vertices of the cyclic polytope, i.e., those linear orderings of the vertices for which Gale’s evenness criterion holds. Proofs give a systematic account on an oriented matroid approach to cyclic polytopes.