Convex hulls of f- and β-vectors

In this paper we describe the convex hulls of the sets of f- and β-vectors of different classes of simplicial complexes on n vertices. These include flag complexes, order complexes of posets, matroid complexes, and general abstract simplicial complexes. As a result of this investigation, standard linear programming problems on these sets can be solved, including maximization of the Euler characteristics or of the sum of the Betti numbers.