The 3^d-conjecture in dimension 4

In 1989 Kalai stated three conjectures concerning centrally symmetric convex polytopes. The weakest of them, the 3^d-conjecture, states that every d-dimensional centrally symmetric polytope has at least 3^d non-empty faces. While the claims in dimensions one, two, and three are, respectively, vacuous, clear, and easy to prove, the 3^d-conjecture in dimension 4 was open until recently. In this talk we present a proof for the 3⁴-conjecture, which relies on a refined lower bound of the flag-functional g₂ on the class of centrally symmetric polytopes. If time permits we also indicate why the other two conjectures are false starting in dimensions 4 and 5, respectively. This is joint work with Axel Werner and Günter M. Ziegler.