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An explicit formula is given for the volume of the polar dual of a polytope. Using this formula, we prove a geometric criterion for critical (w.r.t. volume) sections of a regular simplex.

This paper explores the metrical properties of convex polytopes by means of the classical Pliicker embedding of the Grassmannian G(k, n) of k-planes in R n into the exterior algebra AkR ~. 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.

Abstract. New proofs are given for Cauchy's and Alexandrov's classical theorems on the rigidity of polyhedral frameworks, as well as their higher dimensional generalizations. Through duality, the rigidity of these frameworks follows from characterizations of the case of equality in Minkowski's quadratic inequality.