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Electronic Research Announcements in Mathematical Sciences (ERA-MS)
 

Order isomorphisms in windows

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

 
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Shiri Artstein-Avidan - School of Mathematical Science, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel (email)
Dan Florentin - School of Mathematical Science, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel (email)
Vitali Milman - School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel (email)

Abstract: 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: 26B25.

Received: May 2011;      Revised: July 2011;      Available Online: September 2011.

 References