On the number of faces of centrally-symmetric simplicial polytopes

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 numberf ₀ =2n of vertices. Define integers$h_0 ,...,h_d {\mathbf{ }}by{\mathbf{ }}\sum\limits_{i = 0}^d {f_{i - 1} } (x - 1)^{d - i} = \sum\limits_{i = 0}^d {h_i x^{d - i} }$ A. Björner conjectured (unpublished) that$h_i \geqslant \left( {\begin{array}{*{20}c} d \\ i \\ \end{array} } \right)$ (which generalizes the result of Bárány-Lovász sincef _d−1 =∑ h _i ), and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$h_i - h_{i - 1} \geqslant \left( {\begin{array}{*{20}c} d \\ i \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} d \\ {i - 1} \\ \end{array} } \right),1 \leqslant i \leqslant \left\lfloor {{d \mathord{\left/ {\vphantom {d 2}} \right. \kern-\nulldelimiterspace} 2}} \right\rfloor$, which is easily seen to imply the conjecture of Bárány-Lovász. In this paper the conjectures of Björner are proved.