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

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