Abstract
In this paper we give a characterization of all order isomorphisms on some classes of convex functions. We deal with the class Cvx(K) consisting of lower-semi-continuous convex functions defined on a convex set K, and its subclass \(Cv{x}_{0}(K)\) of non negative functions attaining the value zero at the origin. We show that any order isomorphism on these classes must be induced by a point map on the epi-graphs of the functions, and determine the exact form of this map. To this end we study convexity preserving maps on subsets of \({\mathbb{R}}^{n}\), and also in this area we have some new interpretations, and proofs.
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Acknowledgements
The authors would like to thank Leonid Polterovich for helpful references and comments. They also wish to thank the anonymous referee for useful remarks. Supported in part by Israel Science Foundation: first and second named authors by grant No. 865/07, second and third named authors by grant No. 491/04. All authors were partially supported by BSF grant No. 2006079.
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Artstein-Avidan, S., Florentin, D., Milman, V. (2012). Order Isomorphisms on Convex Functions in Windows. In: Klartag, B., Mendelson, S., Milman, V. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2050. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29849-3_4
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DOI: https://doi.org/10.1007/978-3-642-29849-3_4
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