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Journal of Convex Analysis 28 (2021), No. 3, 751--760
Copyright Heldermann Verlag 2021

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

Dan I. Florentin
Department of Mathematics, Cleveland State University, Cleveland, OH 44115-2214, U.S.A.

Alexander Segal
Afeka Academic College of Engineering, Tel Aviv 69107, Israel


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.

Keywords: Convexity, interpolation, order isomorphisms, duality, Legendre transform, A-transform, J-transform.

MSC: 26B15, 26B25, 26B35, 39B62, 46B06, 52A23, 52A40, 52A41.

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