Vector spaces spanned by the angle sums of polytopes

Kristin A. Camenga

Abstract: This paper describe the spaces spanned by the angle sums of certain classes of polytopes, as recorded in the $\alpha_{}$-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 $\left\lfloor \frac{d-1}{2}\right\rfloor$, 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 $\gamma$-vector, an angle analog to the $h$-vector.

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