Relationships among flag \(f\)-vector inequalities for polytopes
type: Item
instance of: scholarly article
title: Relationships among flag \(f\)-vector inequalities for polytopes
zbMATH Open document ID: 1059.52018
author: Catherine Stenson
published in: Discrete \& Computational Geometry
publication date: 2004-09-22T00:00:00Z
review text: The flag \(f\)-vector of a \(d\)-polytope \(P\) enumerates the numbers of chains \(F_{j(1)}\subset\cdots\subset F_{j(k)}\) of faces of \(P\) with \(\dim F_{j(k)}= j(i)\), for each subset \(S=\{j(1),\dots,j(k)\}\) of \(\{0, \dots,d-1\}\). These data are efficiently encoded in the \textbf{cd}-index of \(P\), which was defined by Jonathan Fine, but first described by \textit{M. Bayer} and \textit{A. Klapper} [Discrete Comput. Geom. 6, No.~1, 33--47 (1991; Zbl 0761.52009)]. For a general \(d\geq 4\), the possibility of classifying completely all the \textbf{cd}-indices of \(d\)-polytopes seems at the moment, somewhat remote; there are not even any plausible conjectures around. Nevertheless, some progress has been made in finding linear inequalities which they satisfy. There are several sources of these. First is the non-negativity of the toric \(g\)-vector, derived from the toric \(h\)-vector of \textit{R. Stanley} [Adv. Stud. Pure Math. 11, 187--213 (1987; Zbl 0652.52007)], now known (in recent work of Kalle Karu) to hold for all polytopes. Next, there is the convolution operation introduced by \textit{G. Kalai} [J. Comb. Theory, Ser. A 49, No. 2, 191--209 (1988; Zbl 0691.05006)], which generates new inequalities from ones for lower-dimensional polytopes. Finally, \textit{L. J. Billera} and \textit{R. Ehrenborg} [Math. Z. 233, 421--441 (2000; Zbl 0966.52014)] showed that the coefficients of the \textbf{cd}-index are minimized by the \(d\)-simplex. In this paper, the author proves a number of interesting results. First, she verifies a conjecture of Meisinger concerning redundancies among the convolutions of the toric 9-vector inequalities. She also shows that these convolutions do not yield the result of Billera and Ehrenborg, and, as a consequence, verifies a conjecture of \textit{G. Meisinger} [Flag numbers and quotients of convex polytopes (Passau: Univ., Fak. f. Math. u. Informatik) (1993; Zbl 0831.52004)] that they do not imply the inequalities \(f_j(P)\geq{d+1\cho…
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 2101565
zbMATH Keywords: polytope
zbMATH Keywords: flag \(f\)-vector
zbMATH Keywords: linear inequalities
zbMATH Keywords: toric \(g\)-vector
zbMATH Keywords: cd-index
zbMATH Keywords: convolution
reviewed by: Peter McMullen
MaRDI profile type: Publication
full work available at URL: s00454-003-0756-0
OpenAlex ID: W2095129581
DOI: 10.1007/S00454-003-0756-0
description: scientific article; zbMATH DE number 2101565