On the Linear Structures Induced by the Four Order Isomorphisms Acting on ${\mathrm{C}\mathrm{v}\mathrm{x}}_0({\mathbb{R}}^n)$

It is known that the volume functional $\varphi \mapsto \int e^{-\varphi }$ satisfies certain concavity or convexity inequalities with respect to three of the four linear structures induced by the order isomorphisms acting on ${\mathrm{C}\mathrm{v}\mathrm{x}}_0({\mathbb{R}}^n)$. In this note we define the fourth linear structure on ${\mathrm{C}\mathrm{v}\mathrm{x}}_0({\mathbb{R}}^n)$ as the pullback of the standard linear structure under the $\mathcal{J}$ 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.