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.
|Journal||Journal of Convex Analysis|
|State||Published - 2021|
Bibliographical noteFunding Information:
Acknowledgements. The authors would like to thank Artem Zvavitch for useful discussions. The first named author was partially supported by the U.S. National Science Foundation Grant DMS-1101636, and also partially supported by the AMS-Simons Travel Grant, which is administered by the American Mathematical Society with support from the Simons Foundation.
© Heldermann Verlag
- Legendre transform
- Order isomorphisms