## On Kalai's conjectures concerning centrally symmetric polytopes

##### Authors
Sanyal, Raman
Werner, Axel
Ziegler, Günter M.
##### Description
In 1989 Kalai stated the three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the $3^d$-conjecture''. It is well-known that the three conjectures hold in dimensions d \leq 3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conjectures B and C fail in all dimensions d \geq 5.
Comment: 14 pages; minor corrections and notational adjustments
##### Keywords
Mathematics - Combinatorics, Mathematics - Metric Geometry, 52B05, 52B12