Raman Sanyal: Cone valuations, Gram's relation, and combinatorics

Abstract

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) among its entries. 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. The interior angle vector of a zonotope is independent of the chosen cone valuation and depends only on the associated lattice of flats. If time permits, we discuss flag-angles as a semi-discrete generalization of flag-vectors and their linear relations. This is joint work with Spencer Backman and Sebastian Manecke.