Order preserving and order reversing operators on the class of convex functions in Banach spaces

TL;DR

This paper extends a known characterization of order preserving and reversing operators from finite-dimensional convex functions to the infinite-dimensional Banach space setting, identifying the structure of such operators.

Contribution

It generalizes the classification of order preserving and reversing operators on convex functions from $ ext{R}^n$ to Banach spaces, revealing their structure in infinite dimensions.

Findings

Order preserving operators are characterized as affine transformations.

Order reversing operators are characterized as Fenchel conjugation.

Results hold for lower semicontinuous proper convex functions in Banach spaces.

Abstract

A remarkable result by S. Artstein-Avidan and V. Milman states that, up to pre-composition with affine operators, addition of affine functionals, and multiplication by positive scalars, the only fully order preserving mapping acting on the class of lower semicontinuous proper convex functions defined on $\mathbb{R}^n$ is the identity operator, and the only fully order reversing one acting on the same set is the Fenchel conjugation. Here fully order preserving (reversing) mappings are understood to be those which preserve (reverse) the pointwise order among convex functions, are invertible, and such that their inverses also preserve (reverse) such order. In this paper we establish a suitable extension of these results to order preserving and order reversing operators acting on the class of lower semicontinous proper convex functions defined on arbitrary infinite dimensional Banach spaces.