  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.
