Extremal Problems Concerning Numbers of Faces

Abstract

In chapters 8 and 9 we obtained information on the f-vectors of d-polytopes, and of some special classes of d-polytopes, by determining the affine hull of all the f-vectors of polytopes in the class considered. In other words, we were concerned with the linear relations satisfied by the f-vectors of all polytopes belonging to a certain class. Chapter 10 reports on the present knowledge about non-linear relations existing between the components of f-vectors of certain classes of polytopes. As will be seen, results in this direction are still rather incomplete—despite the considerable interest and efforts devoted to some aspects of these problems.