## Abstract

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 Cvx_{0}(R^{n}). In this note we define the fourth linear structure on Cvx_{0}(R^{n}) 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.

Original language | English |
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Journal | Journal of Convex Analysis |

Volume | 28 |

Issue number | 3 |

State | Published - 2021 |

### Bibliographical note

Funding 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.

Publisher Copyright:

© Heldermann Verlag

## Keywords

- A-transform
- Convexity
- Duality
- Interpolation
- J-transform
- Legendre transform
- Order isomorphisms

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