# 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.