Valuations on lattice polytopes

Karoly Boroczky, Central European University and Rényi Institute

April 4, 2017

A lattice polytope is the convex hull of finitely many points of the $n$-dimensional integer lattice. A valuation $Z$ on lattice polytopes is a function such that $Z(P\cup Q)+Z(P\cap Q)=Z(P)+Z(Q)$ for any two lattice polytopes $P$ and $Q$ whose union is convex. Typical examples are the lattice point enumerator $G(P)$, the volume $V(P)$ or the difference body $P-P$. In particular, we are mainly interested in valuations that behave "nicely" with respect to translations and the action of the unimodular group.

The talk reviews classical results due to Ehrhart, Betke-Kneser, McMullen, Khovanskii-Pukhilov, then discusses some very recent exciting developments.