Aequationes mathematicae

, Volume 85, Issue 1–2, pp 119–130 | Cite as

Structural results on convexity relative to cost functions

  • Flavia-Corina MitroiEmail author
  • Daniel Alexandru Ion


Mass transportation problems appear in various areas of mathematics, their solutions involving cost convex potentials. Fenchel duality also represents an important concept for a wide variety of optimization problems, both from the theoretical and the computational viewpoints. We drew a parallel to the classical theory of convex functions by investigating the cost convexity and its connections with the usual convexity. We give a generalization of Jensen’s inequality for c-convex functions.

Mathematics Subject Classification



Cost function cost subdifferential cost convex function Jensen inequality Fenchel transform 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Caffarelli L.: Allocation maps with general cost functions. In: Partial Differential Equations and Applications. Lecture Notes in Pure and Appl. Math., vol. 177, pp. 29–35. Dekker, New York (1996)Google Scholar
  2. 2.
    Fenchel W.: On conjugate convex functions. Can. J. Math. 1, 73–77 (1949)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Gangbo W., McCann R.J.: The geometry of optimal transportation. Acta Math. 177, 113–161 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Karakhanyan A., Wang X.-J.: The reflector design problem. Int. Congress Chin. Math. II, 1–4 (2007)Google Scholar
  5. 5.
    Ma X.-N., Trudinger N.S., Wang X.-J.: Regularity of potential functions of the optimal transportation problem. Arch. Ration. Mech. Anal. 177, 151–183 (2005) doi: 10.1007/s00205-005-0362-9 MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Mitroi, F.-C., Niculescu, C.P.: An extension of Young’s inequality. Abstract and Applied Analysis. Article ID 162049. doi: 10.1155/2011/162049
  7. 7.
    Niculescu C.P., Persson L.-E.: Old and new on the Hermite-Hadamard inequality. Real Anal. Exchange 29(2), 663–685 (2004)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Niculescu C.P., Persson L.-E.: Convex Functions and Their Applications. A Contemporary Approach. CMS Books in Mathematics, vol. 23. Springer, New York (2006)Google Scholar
  9. 9.
    Rachev S.T., Rüschendorf L.: Mass Transportation Problems. Probab. Appl. Springer, New York (1998)Google Scholar
  10. 10.
    Rüschendorf L.: Monge-Kantorovich transportation problem and optimal couplings. Jahresber. Deutsch. Math.-Verein. 109(3), 113–137 (2007)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Touchette H., Beck C.: Nonconcave entropies in multifractals and the thermodynamic formalism. J. Stat. Phys. 125, 455–471 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Trudinger N., Wang X.-J.: On strict convexity and continuous differentiability of potential functions in optimal transportation . Arch. Ration. Mech. Anal. 192, 403–418 (2009) doi: 10.1007/s00205-008-0147-z MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Trudinger N., Wang X.-J.: On the second boundary value problem for Monge-Ampère type equations and optimal transportation. Ann. Scuola Norm. Sup. Pisa 8, 1–32 (2009)MathSciNetGoogle Scholar
  14. 14.
    Villani, C.: Optimal Transport. Old and New. Series: Grundlehren der mathematischen Wissenschaften, vol. 338 (2009)Google Scholar
  15. 15.
    Wang X.-J.: On the design of a reflector antenna II. Calc. Var. 20, 329–341 (2004)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CraiovaCraiovaRomania

Personalised recommendations