The bound for the number of faces of a centrally symmetric polytope

For the upper bound for the number of faces of a general polytope, we have Upper bound theorem which says that the cyclic polytope has maximum number of faces given the number of vertices and the dimension of an ambient space. In this talk, I will talk about the case of centrally symmetric polytopes. I will generalize the cyclic polytope to the centrally symmetric case which will tell us a lower bound of the maximum number of faces and talk about upper bound of the maximum number of faces by using a general theory of centrally symmetric polytopes.