Quotient polytopes of cyclic polytopes part I: Structure and characterization

We investigate the quotient polytopesC/F, whereC is a cyclic polytope andF is a face ofC. We describe the combinatorial structure of such quotients, and show that under suitable restrictions the pair (C, F) is determined by the combinatorial type ofC/F. We describe alternative constructions of these quotients by “splitting vertices” of lower-dimensional cyclic polytopes. Using Gale diagrams, we show that every simpliciald-polytope withd+3 vertices is isomorphic to a quotient of a cyclic polytope.