Euler’s Polyhedron Formula and its 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). Similarly, 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 possible to associate an angle to faces of all dimensions and summing angles over faces of fixed dimension gives rise to the interior angle vectors of polytopes. Gram’s relation is the unique linear relation (up to scaling) satisfied by the angle vectors of polytopes. Interestingly, Gram’s relation is independent of the notion of angle. To make this precise, 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, surprisingly, it is the only linear relation, independent of the underlying cone valuation! Behind the scenes a beautiful interplay of discrete geometry and algebraic combinatorics is at work that we will try to explain, starting from the beginning. This is joint work with Spencer Backman and Sebastian Manecke.
Donnerstag, den 28. Juni 2018 um 17.15 Uhr, in INF 205, HS Mathematikon Donnerstag, den 28. Juni 2018 at 17.15, in INF 205, HS Mathematikon
Der Vortrag folgt der Einladung von The lecture takes place at invitation by Frau Dr. G. Battiston