On Kalai's conjectures concerning centrally symmetric polytopes

TL;DR

This paper investigates Kalai's three conjectures on face numbers of centrally symmetric convex polytopes, confirming some conjectures in low dimensions and disproving others in higher dimensions.

Contribution

It provides the first dimension-specific results on Kalai's conjectures, showing which hold or fail in dimensions 4 and above.

Findings

Conjectures A and B hold in dimension 4.

Conjecture C fails in dimension 4.

Both B and C fail in all dimensions d ≥ 5.

Abstract

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.