In 1989, Gil Kalai published three remarkable conjectures about the f-vectors of centrally-symmetric convex polytopes, which he termed Conjectures A, B, and C.

Conjecture A became famous (the 3^d-conjecture): It claims that every centrally-symmetric d-polytope has at least 3^d faces. While this conjecture still stands, and could be proved for dimensions at most 4, Conjectures B and C turned out to be false in dimensions 5 resp. 4 (Sanyal, Werner & Z., 2009).

In the course of subsequent studies, the family of Hansen polytopes derived from certain perfect graphs has come into the focus of investigations. We highlight specific examples that are combinatorially close to Hanner polytopes, with intriguing properties.
(Joint work with Ragnar Freij and Matthias Henze)

Reference:

• Sanyal, Raman ; Werner, Axel ; Ziegler, Günter M. : On Kalai's conjectures concerning centrally symmetric polytopes. Discrete Comput. Geom. 41 (2009), no. 2, p.183-198.