Convex Polytopes: Extremal Constructions and f -Vector Shapes

The study of f -vectors has had huge successes in the last forty years. The most fundamental one is undoubtedly the “g-theorem,” conjectured by McMullen in 1971 and proved by Billera & Lee and Stanley in 1980, which characterizes the f vectors of simplicial and of simple polytopes combinatorially. See also Section 5.2 of Forman’s article in this volume, where h-vectors are discu ssed in connection with the Charney–Davis conjecture. Nevertheless, on some fundamental problems embarassingly little progress was made; one notable such problem concerns the shapes of f -vectors of 4-polytopes. A number of striking and fascinating polytope constructions have been proposed and analyzed over the years. In particular, the Billera–Lee construction produces “all possible f -vectors” of simplicial polytopes. Less visible progress was made outside the range of simple or simplicial polytopes — where our measure of progress is that new polytopes “with interesting f -vectors” should be produced. Thus, still “it seems that overall, we are short of examples. The methods for coming up with useful examples in mathematics (or counterexamples to commonly believed conjectures) are even less clear than the methods for proving mathematical statements” (Gil Kalai, 2000). These lecture notes are meant to display a fruitful interplay of these two areas of study: The discussion of f -vector shapes suggests the notion of “extremal” polytopes, that is, of polytopes with “extremal f -vector shapes.” Our choice of constructions to be discussed here is guided by this: We will be looking at constructions that produce interesting f -vector shapes.