On order preserving and order reversing mappings defined on cones of convex functions

TL;DR

This paper characterizes order reversing and preserving mappings on cones of convex functions in Banach spaces, revealing their structure and resolving open questions through representation theorems involving the Fenchel transform and linear isomorphisms.

Contribution

It provides a complete characterization of order reversing mappings on convex function cones and generalizes the Artstein-Avidan-Milman theorem, addressing previously open problems.

Findings

Fully order reversing mappings exist iff the space is reflexive and isomorphic to its dual.

Order reversing mappings can be represented via the Fenchel transform and a linear isomorphism.

Order preserving mappings on cones of seminorms are characterized by linear isomorphisms.

Abstract

In this paper, we first show that for a Banach space $X$ there is a fully order reversing mapping $T$ from ${\rm conv}(X)$ (the cone of all extended real-valued lower semicontinuous proper convex functions defined on $X$) onto itself if and only if $X$ is reflexive and linearly isomorphic to its dual $X^*$. Then we further prove the following generalized ``Artstein-Avidan-Milman'' representation theorem: For every fully order reversing mapping $T:{\rm conv}(X)\rightarrow {\rm conv}(X)$ there exist a linear isomorphism $U:X\rightarrow X^*$, $x_0^*, \;\varphi_0\in X^*$, $\alpha>0$ and $r_0\in\mathbb R$ so that \begin{equation}\nonumber (Tf)(x)=\alpha(\mathcal Ff)(Ux+x^*_0)+\langle\varphi_0,x\rangle+r_0,\;\;\forall x\in X, \end{equation} where $\mathcal F: {\rm conv}(X)\rightarrow {\rm conv}(X^*)$ is the Fenchel transform. Hence, these resolve two open questions. We also show several representation theorems of fully order preserving mappings defined on certain cones of convex functions. For example, for every fully order preserving mapping $S:{\rm semn}(X)\rightarrow {\rm semn}(X)$ there is a linear isomorphism $U:X\rightarrow X$ so that \begin{equation}\nonumber (Sf)(x)=f(Ux),\;\;\forall f\in{\rm semn}(X),\;x\in X, \end{equation} where ${\rm semn}(X)$ is the cone of all lower semicontinuous seminorms on $X$.