Convex hulls of \(f\)- and \(\beta\)-vectors (Q1380805)

Simplicial complexes play an important rôle in combinatorics, not only in their own right, but as (for instance) flag complexes, order complexes of partially ordered sets, and matroid complexes. Associated with such a complex (on, say, \(n\) vertices) are the numbers \(f_j\) of its \(j\)-cells, and its Betti numbers \(\beta_j\). It is natural to ask for a complete description of the family of all corresponding \(f\)- or \(\beta\)-vectors of complexes in these classes. Less ambitiously, observe that the maxima (or minima) of linear combinations of the \(f_j\) or \(\beta_j\) are determined by the convex hulls of the \(f\)- or \(\beta\)-vectors; among such linear combinations is the Euler characteristic. In this paper, the author determines these convex hulls for each of the classes mentioned at the beginning of the review.