For any set of vectors in , let be the family of integral polytopes whose facet normals are in directions and let be the family of integral polytopes whose edge directions are in directions (H for Hoffman or Hyperplanes, and Ed for Edmond or Edges).
Let , which are edge directions of the simplex with vertices . It is a maximal unimodular system. It is easy to see that (also called convex polytopes or alcoved polytopes) is closed under intersection. The family (also called convex polytopes or generalized polymatroids or generalized permutohedra) is not closed under intersection (e.g. take two pyramid in an octahedron, which is the matroid polytope of that do not share a square face).
But has the property that the intersection of any two (but not three!) polytopes in the the family is an integral polytope. We will call this property (*).
Problem (proven by Isedora Heller): Suppose is a maximal cardinality unimodular system with property (*). Is after a change of lattice basis?
Remark: Let be a graph and the matrix whose columns are for all edges in the graph. We can then construct a graphical matrix and cographical matrix of . Seymour’s decomposition theorem says that every unimodular system is obtained from graphical matrices, cographical matrices, and an exceptional system via two operation , which is putting the matrices (whose columns are the vectors) in a block diagonal form, and is like with an extra row containing a one and the rest zero.
Let be a finite collection of some linear subspaces of . We say that is unimodular if for any subcollection , the group is torsion-free.
Conjecture: Suppose is a unimodular set of vectors such that the lines spanned by vectors in form a maximal unimodular system of subspaces, i.e. we cannot add any other subspaces to the collection while keeping the unimodularity property. Then for some after a change of lattice basis.
Intersect the boundary of the sets ‘s (see the slides) with the set of valuations, which can be assumed to be convex. Call this intersection
Question: Write $late S$ as the intersection of a tropical hyperplane with the set of valuations. Find conditions under which this is possible. How do you find the tropical hyperplane?