On the number of faces of centrally-symmetric simplicial polytopes期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

On the number of faces of centrally-symmetric simplicial polytopes

Authors:	Richard P. Stanley

Affiliation:	(1) Department of Mathematics, Massachusetts Institute of Technology, 02139 Cambridge, MA, USA

Abstract:	I. Bárány and L. Lovász [Acta Math. Acad. Sci. Hung.40, 323–329 (1982)] showed that ad-dimensional centrally-symmetric simplicial polytopeP has at least 2^d facets, and conjectured a lower bound for the numberf_i ofi-dimensional faces ofP in terms ofd and the numberf₀ =2n of vertices. Define integers $h_0 ,...,h_d {mathbf{ }}by{mathbf{ }}sumlimits_{i = 0}^d {f_{i - 1} } (x - 1)^{d - i} = sumlimits_{i = 0}^d {h_i x^{d - i} }$ A. Björner conjectured (unpublished) that $h_i geqslant left( {begin{array}{{20}c} d i end{array} } right)$ (which generalizes the result of Bárány-Lovász sincef_d–1 = h_i), and more strongly that $h_i - h_{i - 1} geqslant left( {begin{array}{{20}c} d i end{array} } right) - left( {begin{array}{*{20}c} d {i - 1} end{array} } right),1 leqslant i leqslant leftlfloor {{d mathord{left/ {vphantom {d 2}} right. kern-nulldelimiterspace} 2}} rightrfloor$ , which is easily seen to imply the conjecture of Bárány-Lovász. In this paper the conjectures of Björner are proved.Partially supported by NSF grant MCS-8104855. The research was performed when the author was a Sherman Fairchild Distinguished Scholar at Caltech.

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