Order isomorphisms in windows

We characterize order preserving transforms on the class of lowersemi- continuous convex functions that are defined on a convex subset of Rn (a "window") and some of its variants. To this end, we investigate convexity preserving maps on subsets of ℝⁿ. 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 Ie{cyrillic, ukrainian} 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.