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On Kalai’s conjectures concerning centrally symmetric polytopes. (English) Zbl 1168.52013
A convex polytope \(P\) is centrally symmetric if \(P = -P\). Let \(f_j\) be the number of \(j\)-dimensional faces of \(P\). G. Kalai [Graphs Comb. 5, No. 1, 389–391 (1989; Zbl 1168.52303)] posed three conjectures on centrally-symmetric \(d\)-dimensional polytopes, of increasing strength:
Conjecture A states that every centrally-symmetric \(d\)-dimensional polytopes has at least \(3^d\) nonempty faces, i.e., \(f_0 + f_1 + \cdots + f_{ d-1 } + 1 \geq 3^d\).
Conjecture B suggests that, given a centrally-symmetric \(d\)-dimensional polytope \(P\), the face numbers of \(P\) are bounded below by the face numbers of a Hanner polytope; Hanner polytopes are built recursively from line segments by taking direct sums or direct products.
Conjecture C strengthens Conjecture B by replacing the bounds on the \(f_j\) by an inequalities involving flag vectors.
It is not hard to see that Kalai’s conjectures hold for \(d \leq 3\). The authors prove that Conjectures A and B for \(d=4\), that Conjecture \(B\) fails for \(d \geq 5\), and that Conjecture C fails for \(d \geq 4\).

52B12 Special polytopes (linear programming, centrally symmetric, etc.)
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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