A centrally symmetric version of the cyclic polytope

Abstract

We define a centrally symmetric analogue of the cyclic polytope and study its facial structure. We conjecture that our polytopes provide asymptotically the largest number of faces in all dimensions among all centrally symmetric polytopes with n vertices of a given even dimension d=2k when d is fixed and n grows. For a fixed even dimension d=2k and an integer 1≤j 0 and at most (1-2-d+o(1)){nj+1} as n grows. We show that c 1(d)≥1-(d-1)-1 and conjecture that the bound is best possible. © 2007 Springer Science+Business Media, LLC.