# American Institute of Mathematical Sciences

2011, 18: 112-118. doi: 10.3934/era.2011.18.112

## Order isomorphisms in windows

 1 School of Mathematical Science, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel, Israel 2 School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978

Received  May 2011 Revised  July 2011 Published  September 2011

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.
Citation: Shiri Artstein-Avidan, Dan Florentin, Vitali Milman. Order isomorphisms in windows. Electronic Research Announcements, 2011, 18: 112-118. doi: 10.3934/era.2011.18.112
##### References:
 [1] S. Artstein-Avidan, D. I. Florentin and V. Milman, "Fractional Linear Maps and Order Isomorphisms for Functions on Windows,", GAFA Lecture Notes, (). [2] S. Artstein-Avidan and V. Milman, The concept of duality in convex analysis, and the characterization of the Legendre transform, Ann. of Math. (2), 169 (2009), 661-674. [3] S. Artstein-Avidan and V. Milman, A characterization of the concept of duality, Electronic Research Announcements in Mathematical Sciences, 14 (2007), 48-65. [4] S. Artstein-Avidan and V. Milman, Hidden structures in the class of convex functions and a new duality transform,, J. Eur. Math. Soc., 13 (). [5] B. Grünbaum, "Convex Polytopes," Second edition, Graduate Texts in Mathematics, 221, Springer-Verlag, New York, 2003. [6] D. Larman, "Recent Results in Convexity," Proceedings of the International Congress of Mathematicians (Helsinki, 1978), 429-434, Acad. Sci. Fennica, Helsinki, 1980. [7] R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. [8] B. Shiffman, Synthetic projective geometry and Poincaré's theorem on automorphisms of the ball, Enseign. Math. (2), 41 (1995), 201-215.

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##### References:
 [1] S. Artstein-Avidan, D. I. Florentin and V. Milman, "Fractional Linear Maps and Order Isomorphisms for Functions on Windows,", GAFA Lecture Notes, (). [2] S. Artstein-Avidan and V. Milman, The concept of duality in convex analysis, and the characterization of the Legendre transform, Ann. of Math. (2), 169 (2009), 661-674. [3] S. Artstein-Avidan and V. Milman, A characterization of the concept of duality, Electronic Research Announcements in Mathematical Sciences, 14 (2007), 48-65. [4] S. Artstein-Avidan and V. Milman, Hidden structures in the class of convex functions and a new duality transform,, J. Eur. Math. Soc., 13 (). [5] B. Grünbaum, "Convex Polytopes," Second edition, Graduate Texts in Mathematics, 221, Springer-Verlag, New York, 2003. [6] D. Larman, "Recent Results in Convexity," Proceedings of the International Congress of Mathematicians (Helsinki, 1978), 429-434, Acad. Sci. Fennica, Helsinki, 1980. [7] R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. [8] B. Shiffman, Synthetic projective geometry and Poincaré's theorem on automorphisms of the ball, Enseign. Math. (2), 41 (1995), 201-215.
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