Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 4 1 2023 01 01 Order isomorphisms and order anti-isomorphisms on spaces of convex functions 27 44 171766 10.30504/jims.2023.385243.1089 EN D. H. Leung Department of Mathematics, National University of Singapore, Singapore, Republic of Singapore. Journal Article 2023 02 09 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. https://jims.ims.ir/article_171766_7e04677f8b346081a3f0bebf4199ffc8.pdf