Subpolytopes of Cyclic Polytopes

A remarkable result of I. Shemer [4] states that the combinatorial structure of a neighbourly 2m-polytope determines the combinatorial structure of each of its subpolytopes. From this, it follows that every subpolytope of a cyclic 2m-polytope is cyclic. In this note, we present a direct proof of this consequence that also yields that certain subpolytopes of a cyclic (2m + 1)-polytope are cyclic. 1 Introduction Let P be a (convex) d-polytope in IE d . The combinatorial structure, or face lattice, of P is the collection of all faces of P ordered by inclusion. We recall that the face lattice of P is completely determined by the set of facets of P , and that two polytopes are combinatorially equivalent if their face lattices are isomorphic. Next, a facet system of P is a pair (F ; X) where X is a finite set, F ` 2 X and there is a bijection f : X \Gamma! vert(P ) such that fconv(ff(v) j v 2 X 0 g) j X 0 2 Fg is the set of facets of P . A subpolytope of P is the convex hull o..