Order isomorphisms and order anti-isomorphisms on spaces of convex functions

Document Type : Dedicated to Prof. A. T.-M. Lau


Department of Mathematics, National University of Singapore, Singapore, Republic of Singapore.


For $i=1,2$, let $C_i$ be a convex set in a locally convex Hausdorff topological vector space $X_i$. Denote by $\operatorname{conv}(C_i)$ the space of all convex, proper, lower semicontinuous functions on $C_i$. A representation is given of any bijection $T:\operatorname{conv}(C_1)\to \operatorname{conv}(C_2)$ that preserves the pointwise order. For $X_i = \mathbb{R}^n$, this recovers a result of Artstein-Avidan and Milman and its generalization by Cheng and Luo. If $X_1$ is a Banach space and $X_2 = X^*_1$ with the weak$^*$-topology, it gives a result due to Iusem, Reem and Svaiter. We also obtain representation of order reversing bijections and thus a characterization of the Legendre transform, generalizing the same result by Artstein-Avidan and Milman for the $\mathbb{R}^n$ case. The result on order isomorphisms actually holds for convex functions with values in ordered topological vector spaces.


Main Subjects

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