Weights on polytopes
The paper refers to the \(g\)-theorem describing the possible face-vectors of simple \(d\)-polytopes \(P\), which was proved by R. P. Stanley using deep techniques of algebraic geometry. Later another approach of the author was given. This proof of the necessity of the conditions of this theorem, while working within the polytope algebra \(\Pi\), in fact only used the spaces of weights of \(P\) which are only subsequently shown actually to belong to \(\Pi\) all along. Here the author uses an independently developed theory of an algebra \(\Omega\) of weights on polytopes to bridge the ``gap''. Since the construction of \(\Omega\) is more approachable than that of \(\Pi\), a yet easier proof of the \(g\)-theorem is obtained. One should mention that the multiplication on weights extends a geometric method of calculating mixed volumes. Indeed, it might well appropriate to think of the weight algebra \(\Omega\) as the algebra of mixed volumes, since here the author also shows that \(\Omega\) is generated as an abelian group by mixed volumes (or even actual volumes) of faces of polytopes. Furthermore, the weight algebra given in the present paper is compared with an alternative one of W. Fulton and B. Sturmfels, where the background is coming from algebraic topology; also, connexions between weights and the concepts of rigidity and stress on polytopes are mentioned.
scientific article; zbMATH DE number 881733