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.
Deep Dive into On Kalais conjectures concerning centrally symmetric polytopes.
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.
A convex d-polytope P is centrally symmetric, or cs for short, if P = -P . Concerning face numbers, this implies that for 0 ≤ i ≤ d -1 the number of i-faces f i (P ) is even and, since P is full-dimensional, that min {f 0 (P ), f d-1 (P )} ≥ 2d. Beyond this, only very little is known for the general case. That is to say, the extra (structural) information of a central symmetry yields no substantial additional constraints for the face numbers on the restricted class of polytopes.
Not uncommon to the f -vector business, the knowledge about face numbers is concentrated on the class of centrally symmetric simplicial, or dually simple, polytopes. In 1982, Bárány and Lovász [3] proved a lower bound on the number of vertices of simple cs polytopes with prescribed number of facets, using a generalization of the Borsuk-Ulam theorem. Moreover, they conjectured lower bounds for all face numbers of this class of polytopes with respect to the number of facets. In 1987 Stanley [24] proved a conjecture of Björner concerning the hvectors of simplicial cs polytopes that implies the one by Bárány and Lovász. The proof uses Stanley-Reisner rings and toric varieties plus a pinch of representation theory. The result of Stanley [24] for cs polytopes was reproved in a more geometric setting by Novik [18] by using “symmetric flips” in McMullen’s weight algebra [16]. For general polytopes, lower bounds on the toric h-vector were recently obtained by A’Campo-Neuen [2] by using combinatorial intersection cohomology. Unfortunately, the toric h-vector contains only limited information about the face numbers of general (cs) polytopes and thus the applicability of the result is limited (see Section 2.1).
In [14], Kalai stated three conjectures about the face numbers of general cs polytopes. Let P be a (cs) d-polytope with f -vector f (P ) = (f 0 , f 1 , . . . , f d-1 ). Define the function s(P ) by s(P ) := 1 + d-1 i=0 f i (P ) = f P (1) where f P (t) := f d-1 (P ) + f d-2 (P )t + • • • + f 0 (P )t d-1 + t d is the f -polynomial. Thus, s(P ) measures the total number of non-empty faces of P . Here is Kalai’s first conjecture from [14], the “3 d -conjecture”. [17]; see also [22].
Conjecture B. For every centrally-symmetric d-polytope P there is a d-dimensional Hanner polytope H such that f i (P ) ≥ f i (H) for all i = 0, . . . , d -1.
for all j = 1, . . . , k. Identifying R 2 [d] with its dual space via the standard inner product, we write α(P ) := S α S f S (P ) for (α S ) S⊆[d] ∈ R 2 [d] . The set
is the polar to the set of flag-vectors of d-polytopes, that is, the cone of all linear functionals that are non-negative on all flag-vectors of (not necessarily cs) d-polytopes.
Conjecture C. For every centrally-symmetric d-polytope P there is a d-dimensional Hanner polytope H such that α(P ) ≥ α(H) for all α ∈ P d .
It is easy to see that C ⇒ B ⇒ A: Define α i (P ) := f i (P ), then α i ∈ P d and the validity of C on the functionals α i implies B; the remaining implication follows since s(P ) is a non-negative combination of the f i (P ).
In this paper we investigate the validity of these three conjectures in various dimensions. Our main results are as follows. The paper is organized as follows. In Section 2 we establish a lower bound on the flag-vector functional g tor 2 on the class of cs 4-polytopes. Together with some combinatorial and geometric reasoning this leads to a proof of Theorem 1.1. In Section 3, we exhibit a centrally symmetric 4-polytope and a flag vector functional that disprove conjecture C. In Section 4 we consider centrally symmetric hypersimplices in odd dimensions; combined with basic properties of Hanner polytopes, this gives a proof of Theorem 1.3. We close with two further interesting examples of centrally symmetric polytopes in Section 5.
Acknowledgements. We are grateful to Gil Kalai for his inspiring conjectures, and for pointing out the connection to symmetric stresses for Theorem 2.1.
In this section we prove Theorem 1.1, that is, the conjectures A and B for polytopes in dimensions d ≤ 4. The work of Stanley [24] implies A and B for simplicial and thus also for simple polytopes. Furthermore, if f 0 (P ) = 2d, then P is linearly isomorphic to a crosspolytope. Therefore, we assume throughout this section that all cs d-polytopes P are neither simple nor simplicial, and that f d-1 (P ) ≥ f 0 (P ) ≥ 2d + 2.
The main work will be in dimension 4. The claims for dimensions one, two, and three are vacuous, clear, and easy to prove, in that order. In particular, the case d = 3 can be obtained from an easy f -vector calculation. But, to get in the right mood, let us sketch a geometric argument. Let P be a cs 3-polytope. Since P is not simplicial, P has a non-triangle facet. Let F be a facet of P with f 0 (F ) ≥ 4 vertices. Let F 0 = P ∩ H with H being the hyperplane parallel to the affine hulls of F and of -F that contains the origin. Now, F 0 is a cs 2-polytope and it is clear that every face G of P that has a
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