Exterior algebra and projections of polytopes

Abstract

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.

A metrical characterization of symmetric polytopes

“…Let Q be a convex polytopr in R 2" with G as its symmetry group. Suppose H < G is a subgroup of G which satisfies the conditions of Theorem 1, and let P = H(Q: L) be the orthogonal projection of Q into the n-space L fixed by H. According to [7,Th. 1], there is a 'projection form' 9 e A~R 2" such that (2.1)…”

The Extreme Projections of the Regular Simplex

“…That is, if P = 11(77": L) and £ e G(k,n) is an orientation of L, then there exists a /c-vector O e Ati?" such that (1.2) fop) = <0,¿:} (see [6,Theorem 1]). This formula holds in a region ;t(<P) c G(k , n) in which the projections have the same combinatorial type as P. In finding extreme projections, we generally work with each region #(<P) separately.…”

“…We shall assume all subspaces belong to //, and that their orientations belong to i(G(k, «)). This restriction is not so important for maximal projections since [6,Proposition 2] shows that the largest projection must lie in H. However, it is crucial for minimal projections since we can choose L e G(k, « + 1 ), L<A\H, with V(Tn : L) as small as desired.…”

“…If G(<&) c / (<P), then the points of C7(<P) represent maximal projections. In general, ||<P|| gives an upper bound on the volumes of projections of 7*" with the same combinatorial type as P. We should mention that Theorem 10 of [6] implies a maximal projection of T" must be simplicial. We begin by calculating the volume of a projection P whose transform P = n(7*": £) , IA e G(n-k,«), is a weighted simplex with weights wx, ... ,wn_k+x , J2w,■ = « + 1 .…”

“…The main tool we shall use in the exterior algebra method of [6], which is summarized in §1. §2 deals with minimal projections, and §3 with maximal projections, where results are given when k or « -k is small (see also [7] and [10]).…”