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On the number of faces of centrally-symmetric simplicial polytopes. (English) Zbl 0611.52002
Author’s abstract: ”I. Bárány and L. Lovász [Acta Math. Acad. Sci. Hung. 40, 323-329 (1982; Zbl 0514.52003)] showed that a d- dimensional centrally-symmetric simplicial polytope \({\mathcal P}\) has at least \(2^ d\) facets, and conjectured a lower bound for the number \(f_ i\) of i-dimensional faces of \({\mathcal P}\) in terms of d and the number \(f_ 0=2n\) of vertices. Define integers \(h_ 0,...,h_ d\) by \(\sum^{d}_{i=0}f_{i-1}(x-1)^{d-i}= \sum^{d}_{i=0}h_ ix^{d-i}.\) A. Björner conjectured (unpublished) that \(h_ i\geq \left( \begin{matrix} d\\ i\end{matrix} \right)\) (which generalizes the result of Bárány-Lovász since \(f_{d-1}=\sum h_ i)\), and more strongly that \(h_ i- h_{i-1}\geq \left( \begin{matrix} d\\ i\end{matrix} \right)- \left( \begin{matrix} d\\ i-1\end{matrix} \right),\) \(1\leq i\leq [d/2],\) which is easily seen to imply the conjecture of Bárány-Lovász. In this paper the conjectures of Björner are proved.”
The proof uses Cohen-Macaulay simplicial complexes and toric varieties. The author points out that for the corresponding upper bound problem (largest possible value of \(f_ i\) for a centrally-symmetric simplicial d-polytope with \(f_ 0=2n\) vertices) not even a plausible conjecture is known.
Reviewer: R.Schneider

MSC:
52Bxx Polytopes and polyhedra
05A20 Combinatorial inequalities
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