# Valuations on Convex Functions

## Monika Ludwig, Technische Universität Wien

## Date and time: 6pm, Tuesday, October 29, 2019

## Place: Courant Institute, WWH1314

A function $\operatorname{\rm Z}$ defined on a space of real-valued
functions ${\mathcal F}$ and taking values in an Abelian semigroup is
called a *valuation* if
$$
\operatorname{\rm Z}(f\vee g)+\operatorname{\rm Z}(f\wedge
g)=\operatorname{\rm Z}(f) +\operatorname{\rm Z} (g)
$$
for all $f,g\in {\mathcal F}$ such that
$f,g, f\vee g, f\wedge g\in {\mathcal F}$. Here $f\vee g$ is the
pointwise maximum of $f$ and $g$, while $f\wedge g$ is their pointwise
minimum. The important, classical notion of valuations on convex,
compact sets is a special case of the rather recent notion of
valuations on function spaces.

We discuss results on valuations defined on various spaces of convex
functions. Classification theorems for SL$(n)$ invariant valuations
and the existence of a homogeneous decomposition for epi-translation
invariant valuations are presented.

Based on joint work with Andrea Colesanti and Fabian Mussnig.