Journal of Geometry

, 109:24 | Cite as

Axiomatic and algebraic convexity of regular pairs



Two dimensional Chebyshev systems, quoted also as regular pairs, induce convex structures both in an axiomatic and in an algebraic way. The aim of this note is to link these structures, by showing that they coincide.


Convex structure Planar convexity Chebyshev system 

Mathematics Subject Classification

Primary 52A10 Secondary 26A51 41A50 52A01 52A35 


  1. 1.
    Baron, K., Matkowski, J., Nikodem, K.: A sandwich with convexity. Math. Pannon. 5(1), 139–144 (1994)MathSciNetMATHGoogle Scholar
  2. 2.
    Barvinok, A.: A Course in Convexity, vol. 54. Graduate Studies in MathematicsAmerican Mathematical Society, Providence (2002)MATHGoogle Scholar
  3. 3.
    Beckenbach, E.F.: Generalized convex functions. Bull. Am. Math. Soc. 43(6), 363–371 (1937)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bessenyei, M., Konkoly, Á., Popovics, B.: Convexity with respect to Beckenbach families. J. Convex Anal. 24(1), 75–92 (2017)MathSciNetMATHGoogle Scholar
  5. 5.
    Bessenyei, M., Páles, Z.S.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003)MathSciNetMATHGoogle Scholar
  6. 6.
    Bessenyei, M., Popovics, B.: Convexity without convex combinations. J. Geom. 107(1), 77–88 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bessenyei, M., Popovics, B.: Convexity structures induces by Chebyshev systems. Indag. Math. (N.S.) 28(6), 1126–1133 (2017)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bessenyei, M., Szokol, P.: Convex separation by regular pairs. J. Geom. 104(1), 45–56 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Carathéodory, C.: Über den variabilitätsbereich der fourierschen konstanten von positiven harmonischen funktionen. Rend. Circ. Mat. Palermo 32, 193–217 (1911)CrossRefMATHGoogle Scholar
  10. 10.
    Karlin, S., Studden, W.J.: Tchebycheff Systems: With Applications in Analysis and Statistics, vol. XV. Pure and Applied Mathematics‘Wiley, New York (1966)MATHGoogle Scholar
  11. 11.
    Krzyszkowski, J.: Generalized convex sets, Rocznik Nauk.-Dydakt. Prace Mat., No. 14, pp. 59–68 (1997)Google Scholar
  12. 12.
    Krzyszkowski, J.: Approximately generalized convex functions. Math. Pannon. 12(1), 93–104 (2001)MathSciNetMATHGoogle Scholar
  13. 13.
    Niculescu, C.P., Persson, L.-E.: Convex Functions and Their Applications. A Contemporary Approach, vol. 23. CMS Books in MathematicsSpringer, New York (2006)CrossRefMATHGoogle Scholar
  14. 14.
    Nikodem, K., Páles, Z.S.: Generalized convexity and separation theorems. J. Convex Anal. 14(2), 239–247 (2007)MathSciNetMATHGoogle Scholar
  15. 15.
    Nikodem, K., Wąsowicz, S.Z.: A sandwich theorem and Hyers-Ulam stability of affine functions, Aequ. Math. 49(1–2), 160–164 (1995)Google Scholar
  16. 16.
    van de Vel, M.L.J.: Theory of Convex Structures, vol. 50. North-Holland Mathematical LibraryNorth-Holland Publishing Co., Amsterdam (1993)MATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of DebrecenDebrecenHungary

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