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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integers $$h_ 0,...,h_ d$$ by $$\sum^{d}_{i=0}f_{i-1}(x-1)^{d-i}= \sum^{d}_{i=0}h_ ix^{d-i}.$$ A. Björner conjectured (unpublished) that $$h_ i\geq \left( \begin{matrix} d\\ i\end{matrix} \right)$$ (which generalizes the result of Bárány-Lovász since $$f_{d-1}=\sum h_ i)$$, and more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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References:
 [1] Bárány, I., Lovász, L.: Borsuk’s theorem and the number of facets of centrally symmetric polytopes. Acta Math. Acad. Sci. Hung.40, 323–329 (1982) · Zbl 0514.52003 · doi:10.1007/BF01903592 [2] Billera, L.J.: Polyhedral theory and commutative algebra. In: Mathematical Programming: The State of the Art, edited by A. Bachem, M. Grötschel, B. Korte, pp. 57–77. Berlin: Springer-Verlag 1986 [3] Danilov, V.I.: The geometry of toric varieties. Russian Math. Surveys33, 97–154 (1978). Translated from Uspekhi Mat. Nauk.33, 85–134 (1978) · Zbl 0425.14013 · doi:10.1070/RM1978v033n02ABEH002305 [4] McMullen, P., Shephard, G.C.: Diagrams for centrally symmetric polytopes. Mathematika15, 123–138 (1968) · Zbl 0167.50902 · doi:10.1112/S0025579300002473 [5] Mebkhout, Z.: Review # 86j: 32027. Math. Reviews 4541 (1986) [6] Saito, M.: Hodge structure via filteredD-modules. Astérisque130, 342–351 (1985) [7] Schneider, R.: Neighbourliness of centrally symmetric polytopes in high dimensions. Mathematika22, 176–181 (1975) · Zbl 0316.52002 · doi:10.1112/S0025579300006045 [8] Stanley, R.: Cohen-Macaulay complexes. In: Higher Combinatorics, edited by M. Aigner, pp. 51–62. Dordrecht-Boston: Reidel 1977 · Zbl 0376.55007 [9] Stanley, R.: Commutative Algebra and Combinatorics. Progress in Mathematics 41. Boston-Basel-Stuttgart: Birkhauser 1983 · Zbl 0537.13009 [10] Stanley, R.: The number of faces of simplicial polytopes and spheres. In: Discrete Geometry and Convexity, Ann. New York Acad. Sci. 440, edited by J.E. Goodman, et al., pp. 212–223. New York: New York Academy of Sciences 1985 [11] Steenbrink, J.H.M.: Mixed Hodge structure on the vanishing cohomology. In: Real and Complex Singularities, Oslo 1976, edited by P. Holm, pp. 525–563. Alphen aan den Rijn: Sijtoff & Noordhoff 1977
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