Order isomorphisms in windows

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

Order isomorphisms in windows

Shiri Artstein-Avidan ^1, , Dan Florentin ^1, and Vitali Milman ^2,

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

Abstract
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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.

Keywords: Convex functions, fractional linear maps, order isomorphisms..

Mathematics Subject Classification: Primary: 52A20; Secondary: 26B2.

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, (). Google Scholar
[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. Google Scholar
[3]	S. Artstein-Avidan and V. Milman, A characterization of the concept of duality,, Electronic Research Announcements in Mathematical Sciences, 14 (2007), 48. Google Scholar
[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 (). Google Scholar
[5]	B. Grünbaum, "Convex Polytopes,", Second edition, 221 (2003). Google Scholar
[6]	D. Larman, "Recent Results in Convexity,", Proceedings of the International Congress of Mathematicians (Helsinki, (1978), 429. Google Scholar
[7]	R. T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970). Google Scholar
[8]	B. Shiffman, Synthetic projective geometry and Poincaré's theorem on automorphisms of the ball,, Enseign. Math. (2), 41 (1995), 201. Google Scholar

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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, (). Google Scholar
[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. Google Scholar
[3]	S. Artstein-Avidan and V. Milman, A characterization of the concept of duality,, Electronic Research Announcements in Mathematical Sciences, 14 (2007), 48. Google Scholar
[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 (). Google Scholar
[5]	B. Grünbaum, "Convex Polytopes,", Second edition, 221 (2003). Google Scholar
[6]	D. Larman, "Recent Results in Convexity,", Proceedings of the International Congress of Mathematicians (Helsinki, (1978), 429. Google Scholar
[7]	R. T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970). Google Scholar
[8]	B. Shiffman, Synthetic projective geometry and Poincaré's theorem on automorphisms of the ball,, Enseign. Math. (2), 41 (1995), 201. Google Scholar

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