Vector spaces spanned by the angle sums of polytopes
Abstract
Description
In this paper, we will describe the space spanned by the angle-sums of polytopes, recorded in the alpha-vector. We will consider the angles sums of simplices and the angles sums and face numbers of simplicial polytopes and general polytopes. We will construct families of polytopes whose angle sums span the spaces of polytopes defined by the Gram and Perles equations, analogues of the Euler and Dehn-Sommerville equations. We show that the dimensions of the affine span of the space of angle sums of simplices is floor[(d-1)/2] + 1, and that of the angle sums and face numbers of simplicial polytopes and general polytopes are d-1 and 2d-3 respectively.
16 pages, 1 figure, submitted to Beitrage zur Algebra und Geometrie v. 2, addition of section on angle analog of h-vector and changes in proofs of affine independence
16 pages, 1 figure, submitted to Beitrage zur Algebra und Geometrie v. 2, addition of section on angle analog of h-vector and changes in proofs of affine independence