Exterior algebra and projections of polytopes

Abstract

This paper explores the metrical properties of convex polytopes by means of the classical Plücker embedding of the Grassmannian G(k, n) of k-planes in Rn into the exterior algebra ΛkRn. The results follow from the description of the volume of the projection of a polytope into a k-plane by a piecewise linear function on G(k, n). For example, the Hodge-star operator is used to obtain the volume of a polytope from its Gale transform. Also, the classification of the faces of G(2, n) (or G(n-2, n)) imply that the largest projection within a particular combinatorial type is unique if k=2 or n-2. © 1990 Springer-Verlag New York Inc.