Convex Polytopes: Extremal Constructions and f -Vector Shapes

These lecture notes treat some current aspects of two closely interrelated topics from the theory of convex polytopes: the shapes of f -vectors, and extremal constructions. 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 ?? of Forman’s article in this volume, where h-vectors are discussed 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 has 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