On simple polytopes

Abstract

LetP be a simpled-polytope ind-dimensional euclidean space $$\mathbb{E}^d $$ , and let Π(P) be the subalgebra of the polytope algebra Π generated by the classes of summands ofP. It is shown that the dimensions of the weight spacesΞ r(P) of Π(P) are theh-numbers ofP, which describe the Dehn-Sommerville equations between the numbers of faces ofP, and reflect the duality betweenΞ r (P) andΞ d-r (P). Moreover, Π(P) admits a Lefschetz decomposition under multiplication by the element ofΞ 1(P) corresponding toP itself, which yields a proof of the necessity of McMullen's conditions in theg-theorem on thef-vectors of simple polytopes. The Lefschetz decomposition is closely connected with the new Hodge-Riemann-Minkowski quadratic inequalities between mixed volumes, which generalize Minkowski's second inequality; also proved are analogous generalizations of the Aleksandrov-Fenchel inequalities. A striking feature is that these are obtained without using Brunn-Minkowski theory; indeed, the Brunn-Minkowski theorem (without characterization of the cases of equality) can be deduced from them. The connexion found between Π(P) and the face ring of the dual simplicial polytopeP * enables this ring to be looked at in two ways, and a conjectured formulation of theg-theorem in terms of a Gale diagram ofP * is also established.