Order Isomorphisms on Convex Functions in Windows

  • Shiri Artstein-AvidanEmail author
  • Dan Florentin
  • Vitali Milman
Part of the Lecture Notes in Mathematics book series (LNM, volume 2050)


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.


Order Isomorphism Point Map Fractional Linear Map Conservative Interval Order Preservation 
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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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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Shiri Artstein-Avidan
    • 1
    Email author
  • Dan Florentin
    • 1
  • Vitali Milman
    • 1
  1. 1.Sackler Faculty of Exact Sciences, Department of MathematicsTel Aviv UniversityTel AvivIsrael

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