**Gleb Koshevoy**

### Good Exercise:

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.

### Question:

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.

## John Weymark

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?