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Journal of Geometry

, Volume 104, Issue 1, pp 45–56 | Cite as

Convex separation by regular pairs

  • Mihály BessenyeiEmail author
  • Patrícia Szokol
Article

Abstract

It is known that the existence of a convex (resp., concave) separator between two given functions can be characterized via a simple inequality. The notion of convexity can be generalized applying regular pairs (in other words, two dimensional Chebyshev systems). The aim of the present note is to extend the above mentioned result to this setting. In the proof, a modified version of the classical Carathéodory’s theorem and the characterization of convex functions play the key role.

Mathematics Subject Classification (2010)

26A51 39B62 52A20 

Keywords

Convexity Chebyshev systems separation theorems Carathéodory’s theorem 

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References

  1. 1.
    Balaj M., Wa¸sowicz Sz.: Haar spaces and polynomial selections. Math Pannon. 14, 63–70 (2003)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Baron K., Matkowski J., Nikodem K.: A sandwich with convexity. Math. Pannon. 5, 139–144 (1994)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bessenyei, M.: Hermite-Hadamard-type inequalities for generalized convex functions. JIPAM. J. Inequal. Pure Appl. Math. 9, Article 63, p. 51 (2008)Google Scholar
  4. 4.
    Bessenyei M., Páles Zs.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6, 379–392 (2003)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bessenyei M., Páles Zs.: Separation by linear interpolation families. J. Nonlinear Convex Anal. 13, 49–56 (2012)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bonsall, F.F.: The characterization of generalized convex functions. Quart. J. Math. Oxford Ser. (2) 1, 100–111 (1950)Google Scholar
  7. 7.
    Hyers D.H., Ulam S.M.: Approximately convex functions. Proc. Am. Math. Soc. 3, 821–828 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Házy A., Páles Zs.: On approximately midconvex functions. Bull. London Math. Soc. 36, 339–350 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Karlin, S., Studden, W.J.: Tchebycheff Systems: With Applications in Analysis and Statistics. Pure and Applied Mathematics, vol. XV, Interscience, New York (1966)Google Scholar
  10. 10.
    Krzyszkowski J.: Generalized convex sets. Rocznyk Nauk.-Dydaktik Prace Mat. 14, 59–68 (1997)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Krzyszkowski J.: Approximately generalized convex functions. Math. Pannon. 12, 93–104 (2001)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Lay, S.R.: Convex Sets and Their Applications. Wiley-Interscience New York (1982)Google Scholar
  13. 13.
    Magaril-Il’yaev, G.G., Tikhomirov, V.M.: Convex Analysis: Theory and Applications. American Mathematical Society, Providence (2003)Google Scholar
  14. 14.
    Niculescu, C.P., Persson L.-E.: Convex Functions and Their Applications. CMS Books in Mathematics, vol. 23, Springer, New York (2006)Google Scholar
  15. 15.
    Nikodem K., Páles Zs.: Generalized convexity and separation theorems. J. Convex Anal. 14, 239–248 (2007)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Nikodem K., Wa¸sowicz Sz.: A sandwich theorem and Hyers-Ulam stability of affine functions. Aequationes Math. 49, 160–164 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Pólya Gy.: On the mean-value theorem corresponding to a given linear homogeneous differential equation. Trans. Am. Math. Soc. 24, 312–324 (1922)CrossRefGoogle Scholar
  18. 18.
    Pólya Gy.: Untersuchungen über Lücken und Singularitäten von Potenzreihen. Math. Z. 29, 549–640 (1929)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Páles Zs.: On approximately convex functions. Proc. Am. Math. Soc. 131, 243–252 (2003)zbMATHCrossRefGoogle Scholar
  20. 20.
    Wa¸sowicz Sz.: Polynomial selections and separation by polynomials. Studia Math. 120, 75–82 (1996)MathSciNetGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of DebrecenDebrecenHungary

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