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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 $$d \geq 4$$.

##### MSC:
 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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