From valuations on convex bodies to convex functions

TL;DR

This paper develops a geometric framework connecting valuations on convex bodies and functions, using classical and new characterizations to advance understanding of valuation properties.

Contribution

It introduces a framework linking valuations on convex bodies and functions, and provides new characterizations of certain valuation classes using classical results.

Findings

Characterization of continuous, epi-translation invariant, n-epi-homogeneous valuations on convex functions.

New approach to 1-epi-homogeneous valuations based on Goodey and Weil's method.

Extension of classical valuation results to the setting of convex functions.

Abstract

A geometric framework relating valuations on convex bodies to valuations on convex functions is introduced. It is shown that a classical result by McMullen can be used to obtain a characterization of continuous, epi-translation invariant, and n-epi-homogeneous valuations on convex functions, which was previously established by Colesanti, Ludwig, and Mussnig. Following an approach by Goodey and Weil, a new characterization of 1-epi-homogeneous valuations is obtained.