Skip to main content

Stability of Order Preserving Transforms

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. S. Artstein-Avidan, V. Milman, A new duality transform, C. R. Acad. Sci. Paris 346, 1143–1148 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. S. Artstein-Avidan, V. Milman, The concept of duality for measure projections of convex bodies. J. Funct. Anal. 254, 2648–2666 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  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)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. S. Artstein-Avidan, V. Milman, Hidden structures in the class of convex functions and a new duality transform. J. Eur. Math. Soc. 13, 975–1004

    Google Scholar 

  5. S. Artstein-Avidan, V. Milman, Stability results for some classical convexity operations. To appear in Advances in Geometry

    Google Scholar 

  6. D.H. Hyers, On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A. 27, 222–224 (1941)

    CrossRef  MathSciNet  Google Scholar 

  7. S.S. Kutateladze, A.M. Rubinov, The Minkowski Duality and Its Applications (Russian) (Nauka, Novosibirsk, 1976)

    Google Scholar 

  8. R. Schneider, The endomorphisms of the lattice of closed convex cones. Beitr. Algebra Geom. 49, 541–547 (2008)

    MATH  Google Scholar 

  9. S.M. Ulam, A Collection of Mathematical Problems (Interscience Publ., New York, 1960)

    MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dan Florentin .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Florentin, D., Segal, A. (2012). Stability of Order Preserving Transforms. In: Klartag, B., Mendelson, S., Milman, V. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2050. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29849-3_12

Download citation