Order Isomorphisms on Convex Functions in Windows

TL;DR

This paper characterizes all order isomorphisms on classes of convex functions, showing they are induced by point maps on epi-graphs, with new insights into convexity-preserving maps.

Contribution

It provides a complete characterization of order isomorphisms on convex function classes and introduces new interpretations and proofs for convexity-preserving maps.

Findings

Order isomorphisms are induced by point maps on epi-graphs

Explicit form of the inducing point map is determined

New interpretations and proofs for convexity-preserving maps

Abstract

In this paper we give a characterization of all order isomorphisms on some classes of convex functions. We deal with the class $Cvx(K)$ consisting of lower-semi-continuous convex functions defined on a convex set $K$, and its subclass $Cvx_0(K)$ of non negative functions attaining the value zero at the origin. We show that any order isomorphism on these classes must be induced by a point map on the epi-graphs of the functions, and determine the exact form of this map. To this end we study convexity preserving maps on subsets of ${\mathbb R}^n$, and also in this area we have some new interpretations, and proofs.