On the Linear Structures Induced by the Four Order Isomorphisms Acting on ${\rm{Cvx}}_0({\mathbb R}^n)$

TL;DR

This paper explores the linear structures induced by four order isomorphisms on convex functions, revealing that the newly defined structure does not support concavity or convexity inequalities and analyzing their order relations.

Contribution

It introduces a new linear structure on convex functions via the ${ m{f J}}$ transform and studies its properties and relations with existing structures.

Findings

The new linear structure does not satisfy concavity or convexity inequalities.

A quasi-convexity inequality is only violated by a factor of 2.

All order relations among the four structures are characterized.

Abstract

It is known that the volume functional $\,\phi\mapsto\int e^{-\phi}\,$ satisfies certain concavity or convexity inequalities with respect to three of the four linear structures induced by the order isomorphisms acting on ${\rm{Cvx}}_0({\mathbb R}^n)$. In this note we define the fourth linear structure on ${\rm{Cvx}}_0({\mathbb R}^n)$ as the pullback of the standard linear structure under the ${\cal J}$ transform. We show that, interpolating with respect to this linear structure, no concavity or convexity inequalities hold, and prove that a quasi-convexity inequality is violated only by up to a factor of $2$. We also establish all the order relations which the four different interpolations satisfy.