Vector Spaces Spanned by the Angle Sums of Polytopes

This paper describe the spaces spanned by the angle sums of certain classes of polytopes, as recorded in the α-vector. Families of polytopes are constructed whose angle sums span the spaces of polytopes defined by the Gram and Perles equations, analogs of the Euler and Dehn-Sommerville equations. This shows that the dimension of the affine span of the space of angle sums of simplices is ⌊ d−1 2 ⌋ , and that of the combined angle sums and face numbers of simplicial polytopes and general polytopes are d− 1 and 2d− 3, respectively. A tool used in proving these results is the γ-vector, an angle analog to the h-vector.