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

D. I. Florentin ; A. Segal

Journal of convex analysis, Tome 28 (2021) no. 3, pp. 751-76

Résumé

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.

Détail

Classification : 26B15, 26B25, 26B35, 39B62, 46B06, 52A23, 52A40, 52A41
Mots-clés : Convexity, interpolation, order isomorphisms, duality, Legendre transform, A-transform, J-transform