Cyclic Polytopes and Oriented Matroids

Consider the moment curve in the real Euclidean space R defined parametrically by the map γ : R → R, t → γ(t) = (t, t, . . . , t). The cyclic d-polytope Cd(t1, . . . , tn) is the convex hull of the n, n > d, different points on this curve. The matroidal analogues are the alternating oriented uniform matroids. A polytope [resp. matroid polytope] is called cyclic if its face lattice is isomorphic to that of Cd(t1, . . . , tn). We give combinatorial and geometrical characterizations of cyclic [matroid] polytopes. A simple evenness criterion determining the facets of Cd(t1, . . . , tn) was given by David 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. ∗1991 Mathematics Subject Classification: Primary 05B35, 52A25.