Equivariant Valuations on Convex Functions

TL;DR

This paper classifies all continuous valuations on convex functions that are invariant under certain transformations, identifying analogues of classical geometric maps and proving the non-existence of a projection body analogue.

Contribution

It provides a complete classification of specific valuations on convex functions, revealing the absence of a projection body type map in this functional setting.

Findings

Identified valuation-theoretic analogues of the difference body map.

Proved the non-existence of a generalization of the projection body map.

Classified all continuous, volume-preserving, duality-invariant valuations.

Abstract

We classify all continuous valuations on the space of finite convex functions with values in the same space which are dually epi-translation-invariant and equi- resp. contravariant with respect to volume-preserving linear maps. We thereby identify the valuation-theoretic functional analogues of the difference body map and show that there does not exist a generalization of the projection body map in this setting. This non-existence result is shown to also hold true for valuations with values in the space of convex functions that are finite in a neighborhood of the origin.