A homogeneous decomposition theorem for valuations on convex functions

TL;DR

This paper proves a homogeneous decomposition theorem for valuations on convex functions, classifies certain valuations, and extends results via duality to finite-valued convex functions.

Contribution

It establishes a homogeneous decomposition theorem for valuations on convex functions and classifies epi-homogeneous valuations, extending results through duality.

Findings

Existence of a homogeneous decomposition for continuous, epi-translation invariant valuations.

Classification of epi-homogeneous valuations of degree n.

Duality results for valuations on finite-valued convex functions.

Abstract

The existence of a homogeneous decomposition for continuous and epi-translation invariant valuations on super-coercive functions is established. Continuous and epi-translation invariant valuations that are epi-homogeneous of degree $n$ are classified. By duality, corresponding results are obtained for valuations on finite-valued convex functions.