# Gram’s relation, cone valuations, and combinatorics

Raman Sanyal, Goethe-University Frankfurt
PDL C-401

Euler’s Polyhedron Formula and it’s generalization, the Euler-Poincare formula, is a cornerstone of the combinatorial theory of polytopes. It states that the number of faces of various dimensions of a convex polytope satisfy a linear relation and it is the only linear relation (up to scaling). Gram’s relation generalizes the fact that the sum of (interior) angles at the vertices of a convex \$n\$-gon is \$(n-2)\pi\$. In dimensions \$3\$ and up, it is necessary to consider angles at all faces. This gives rise to the interior angle vector of a convex polytope and Gram’s relation is the unique linear relation (up to scaling). In this talk, we will consider generalizations of “angles” in the form of cone valuations. It turns out that the associated generalized angle vectors still satisfy Gram’s relation and that it is the only linear relation, independent of the notion of “angle”. To prove such a result, we rely on a very powerful connection to the combinatorics of zonotopes and hyperplane arrangements. This is joint work with Spencer Backman and Sebastian Manecke.

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