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In this paper, we first show that for a Banach space $X$ there is a fullyorder reversing mapping $T$ from ${\rm conv}(X)$ (the cone of all extendedreal-valued lower semicontinuous proper convex functions defined on $X$) ontoitself if and only if $X$ is reflexive and linearly isomorphic to its dual$X^*$. Then we further prove the following generalized``Artstein-Avidan-Milman'' representation theorem: For every fully orderreversing mapping $T:{\rm conv}(X)\rightarrow {\rm conv}(X)$ there exist alinear isomorphism $U:X\rightarrow X^*$, $x_0^*, \;\varphi_0\in X^*$,$\alpha>0$ and $r_0\in\mathbb R$ so that \begin{equation}\nonumber(Tf)(x)=\alpha(\mathcal Ff)(Ux+x^*_0)+\langle\varphi_0,x\rangle+r_0,\;\;\forallx\in X, \end{equation} where $\mathcal F: {\rm conv}(X)\rightarrow {\rmconv}(X^*)$ is the Fenchel transform. Hence, these resolve two open questions.We also show several representation theorems of fully order preserving mappingsdefined on certain cones of convex functions. For example, for every fullyorder preserving mapping $S:{\rm semn}(X)\rightarrow {\rm semn}(X)$ there is alinear isomorphism $U:X\rightarrow X$ so that \begin{equation}\nonumber(Sf)(x)=f(Ux),\;\;\forall f\in{\rm semn}(X),\;x\in X, \end{equation} where${\rm semn}(X)$ is the cone of all lower semicontinuous seminorms on $X$.

On order preserving and order reversing mappings defined on cones of convex functions