Monotone valuations on the space of convex functions

TL;DR

This paper investigates valuations on convex functions in Euclidean space, providing integral representations for those that are monotone, invariant under rigid motions, and satisfy certain continuity, with special focus on simple and homogeneous valuations.

Contribution

It introduces new integral representation formulas for monotone, rigid motion invariant valuations on convex functions, especially for simple and homogeneous cases.

Findings

Integral representation formulas derived for specific valuations

Characterization of valuations invariant under rigid motions

Results applicable to simple and homogeneous valuations

Abstract

We consider the space of convex functions defined in the Euclidean $n$-dimensional space, which are lower semi-continuous and tend to infinity at infinity. We study real-valued valuations defined on this space of functions, which are invariant under the composition with rigid motions, monotone and verify a certain type of continuity. Among these valuations we prove integral representation formulas for those which are, additionally, simple or homogeneous.