Relationships among flag \(f\)-vector inequalities for polytopes
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 \(f_j(P)\geq{d+1\choose j+1}\). Conversely, she proves that not all the toric \(g\)-vector inequalities are implied by the \textbf{cd}-index inequalities, and that there are also redundancies among these inequalities. Finally, she shows that convolutions of \textbf{cd} coefficients are just nonnegative linear combinations of other \textbf{cd} coefficients, so that convolution yields no new non-negativity conditions.
scientific article; zbMATH DE number 2101565