Order isomorphisms in windows

We characterize order preserving transforms on the class of lower-semi-continuous convex functions that are defined on a convex subset of $\mathbb{R}^n$ (a "window") and some of its variants. To this end, we investigate convexity preserving maps on subsets of $\mathbb{R}^n$. We prove that, in general, an order isomorphism is induced by a special convexity preserving point map on the epi-graph of the function. In the case of non-negative convex functions on $K$, where $0\in K$ and $f(0) = 0$, one may naturally partition the set of order isomorphisms into two classes; we explain the main ideas behind these results.