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.