On the Linear Structures Induced by the Four Order Isomorphisms Acting on Cvx_(0)(R^(n))

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.