Face Numbers of Centrally Symmetric Polytopes Produced from Split Graphs.

We analyze a remarkable class of centrally symmetric polytopes, the Hansen polytopes of split graphs. We confirm Kalai's 3(d) conjecture for such polytopes (they all have at least 3(d) nonempty faces) and show that the Hanner polytopes among them (which have exactly 3(d) nonempty faces) correspond to threshold graphs. Our study produces a new family of Hansen polytopes that have only 3(d) + 16 nonempty faces.