On Kalai's conjectures concerning centrally symmetric polytopes

Details On Kalai's conjectures concerning centrally symmetric polytopes On Kalai's conjectures concerning centrally symmetric polytopes

2007-08-27

arxiv.org/abs/0708.3661v2

Mathematics

Combinatorics

14 pages; minor corrections and notational adjustments

Scientific paper

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.

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