Type Fans of Alcoved Polytopes, Nestohedra, and Simple Games

The support cone is the closure of the set of polyhedra with a given set of facet directions. Support cones have a natural subdivision into type cones, that is, into regions with fixed normal fan. Support cones and their type fans arise in various contexts ranging from convex geometry to parametric optimization but, much like secondary fans, are notoriously difficult to compute.

Alcoved polyhedra are (lattice) polyhedra that are unions of alcoves of the affine reflection group of type A. Every alcoved polyhedron can be uniquely described by an edge-weighted directed graph. In this talk, I discuss support cones of alcoved polyhedra and their type fans from a combinatorial perspective.

I focus on two classes of examples. For rooted tree posets, the corresponding type fans are described by certain nestohedra. For double cycles, the type fans are related to weighted simple (voting) games from cooperative game theory. This is joint work with Aenne Benjes and Benjamin Schröter.