On Kalai’s Conjectures Concerning Centrally Symmetric Polytopes

In 1989, Kalai stated three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the “3d-conjecture.” It is well known that the three conjectures hold in dimensions d≤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≥5.

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