On the Linear Structures Induced by the Four Order Isomorphisms Acting on Cvx₀(Rⁿ)

It is known that the volume functional ϕ 7→ ∫ e^−ϕ satisfies certain concavity or convexity inequalities with respect to three of the four linear structures induced by the order isomorphisms acting on Cvx₀(Rⁿ). In this note we define the fourth linear structure on Cvx₀(Rⁿ) as the pullback of the standard linear structure under the 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.