# Katharina Jochemko: Generalized permutahedra: Minkowski linear functionals and Ehrhart positivity

**Time: **
Wed 2019-10-09 10.15 - 11.00

**Location: **
Room 3418, Lindstedtsvägen 25. Department of Mathematics, KTH

**Participating: **
Katharina Jochemko

Abstract: Generalized permutahedra form a combinatorially rich class of polytopes

that naturally appear in many areas of mathematics such as

combinatorics, geometry, optimization and statistics. They comprise many

important classes of polytopes, for example, matroid polytopes. We study

functions on generalized permutahedra that behave linearly with respect

to dilation and taking Minkowski sums. We give a complete classification

of all positive, translation-invariant Minkowski linear functionals on

permutahedra that are invariant under permutations of the coordinates:

they form a simplicial cone and we explicitly describe the generators.

We apply our results to prove that the linear coefficients of Ehrhart

polynomials of generalized permutahedra are nonnegative, verifying

conjectures of De Loera-Haws-Koeppe (2009) and Castillo-Liu

(2018) in this case. This is joint work with Mohan Ravichandran.