Quotient polytopes of cyclic polytopes part I: Structure and characterization

We investigate the quotient polytopes C/F, where C is a cyclic polytope and F is a face of C. We describe the combinatorial structure of such quotients, and show that under suitable restrictions the pair (C, F) is determined by the combinatorial type of C/F. We describe alternative constructions of these quotients by "splitting vertices" of lower-dimensional cyclic polytopes. Using Gale diagrams, we show that every simplicial d-polytope with d+3 vertices is isomorphic to a quotient of a cyclic polytope.