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Stability of Order Preserving Transforms

  • Dan FlorentinEmail author
  • Alexander Segal
Chapter
  • 1.8k Downloads
Part of the Lecture Notes in Mathematics book series (LNM, volume 2050)

Abstract

The purpose of this paper is to show stability of order preserving/ reversing transforms on the class of non-negative convex functions in \({\mathbb{R}}^{n}\), and its subclass, the class of non-negative convex functions attaining 0 at the origin (these are called “geometric convex functions”). We show that transforms that satisfy conditions which are weaker than order preserving transforms, are essentially close to the order preserving transforms on the mentioned structures.

Keywords

Order Preservation Convex Indicator Function Extreme Family Convex Bodies Maximum Triangle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The authors would like to express their sincere appreciation to Prof. Vitali Milman and Prof. Shiri Artstein-Avidan for their support, advice and discussions. Dan Florentin was Partially supported by the Israel Science Foundation Grant 865/07. Alexander Segal was Partially supported by the Israel Science Foundation Grant 387/09.

References

  1. 1.
    S. Artstein-Avidan, V. Milman, A new duality transform, C. R. Acad. Sci. Paris 346, 1143–1148 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    S. Artstein-Avidan, V. Milman, The concept of duality for measure projections of convex bodies. J. Funct. Anal. 254, 2648–2666 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    S. Artstein-Avidan, V. Milman, The concept of duality in asymptotic geometric analysis, and the characterization of the Legendre transform. Ann. Math. 169(2), 661–674 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    S. Artstein-Avidan, V. Milman, Hidden structures in the class of convex functions and a new duality transform. J. Eur. Math. Soc. 13, 975–1004Google Scholar
  5. 5.
    S. Artstein-Avidan, V. Milman, Stability results for some classical convexity operations. To appear in Advances in GeometryGoogle Scholar
  6. 6.
    D.H. Hyers, On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A. 27, 222–224 (1941)MathSciNetCrossRefGoogle Scholar
  7. 7.
    S.S. Kutateladze, A.M. Rubinov, The Minkowski Duality and Its Applications (Russian) (Nauka, Novosibirsk, 1976)Google Scholar
  8. 8.
    R. Schneider, The endomorphisms of the lattice of closed convex cones. Beitr. Algebra Geom. 49, 541–547 (2008)zbMATHGoogle Scholar
  9. 9.
    S.M. Ulam, A Collection of Mathematical Problems (Interscience Publ., New York, 1960)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.School of Mathematical Science, Sackler Faculty of Exact ScienceTel Aviv UniversityTel AvivIsrael

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