## Invited Session Fri.3.H 2032

#### Friday, 15:15 - 16:45 h, Room: H 2032

**Cluster 11: Integer & mixed-integer programming** [...]

### Integer points in polytopes II

**Chair: Michael Joswig and Günter M. Ziegler**

**Friday, 15:15 - 15:40 h, Room: H 2032, Talk 1**

**Benjamin Nill**

Recent developments in the geometry of numbers of lattice polytopes

**Abstract:**

In this talk, I will give an overview about recent results in the geometry of numbers of lattice polytopes. All of these will deal with the question of what we know about lattice polytopes with a certain number of interior lattice points or none at all. I also hope to show how an invariant in Ehrhart theory

possibly allows a unifying view on these results.

**Friday, 15:45 - 16:10 h, Room: H 2032, Talk 2**

**Andreas Paffenholz**

Permutation polytopes

**Coauthors: Barbara Baumeister, Christian Haase, Benjamin Nill**

**Abstract:**

A permutation polytope is the convex hull of the permutation matrices of a subgroup of *S*_{n}. These polytopes are a special class of *0/1*-polytopes. A well-known example is the Birkhoff polytope of all doubly-stochastic matrices defined by the symmetric group *S*_{n}. This is a well studied polytope.

Much less is known about general permutation polytopes. I will shortly discuss basic properties, combinatorial

characterizations, lattice properties, and connections between the group and the polytope.

A main focus of my presentation will be on recent results for cyclic groups. Their permutation polytopes correspond to marginal polytopes studied in algebraic statistics and optimization. In particular, I will present families of facet defining inequalities.

**Friday, 16:15 - 16:40 h, Room: H 2032, Talk 3**

**Alexander Mieczyslaw Kasprzyk**

Riemannian polytopes

**Coauthor: Gabor Hegedus**

**Abstract:**

Given a convex lattice polytope *P*, one can count the number of points in a dilation *mP* via the Ehrhart polynomial *L*_{P}. The roots of *L*_{P} (over **C**) have recently been the subject of much study, with a particular focus on the distribution of the real parts. In particular, V. Golyshev conjectured, and the authors recently proved, that any smooth polytope of dimension at most five are so-called Riemannian polytopes; this is, the roots of *L*_{P} all satisfy * ℜ (z)=-1/2*.

I shall discuss some recent results on Riemannian polytopes, with particular emphasis on reflexive polytopes. In particular, I will discuss the distribution of the roots in the case of a reflexive polytope *P*, and a characterisation of when *P* is Reimannian.