How many cyclic subpolytopes can a non-cyclic polytope have? (Q1062282)

A d-dimensional polytope is neighborly if every [d/2] of its vertices determine a facet, where [x] is the integer part of x. If \(v>d\) distinct points are selected on the moment curve \(\{(t,t^ 2,...,t^ d);\quad t\in {\mathbb{R}}\},\) their convex hull is a neighborly polytope C(v,d). Any polytope combinatorially equivalent to some C(v,d) is called a cyclic d- polytope. A subpolytope of a polytope P is the convex hull of a subset of the vertices of P. The author proves that a d-polytope with v vertices which is not cyclic has at most \(d+1\) cyclic d-dimensional subpolytopes with v-1 vertices if d is even and \(v\geq d+5.\)