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

Dan I. Florentin; Alexander Segal

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

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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 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.

Original language	English
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

Cite this

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

Abstract

Bibliographical note

Keywords

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