Skip to main content

Advertisement

SpringerLink
Log in

Jensen’s inequality on convex spaces

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We prove a Jensen’s inequality on \(p\)-uniformly convex space in terms of \(p\)-barycenters of probability measures with \((p-1)\)-th moment with \(p\in ]1,\infty [\) under a geometric condition, which extends the results in Kuwae (Jensen’s inequality over CAT\((\kappa )\)-space with small diameter. In: Proceedings of Potential Theory and Stochastics, Albac Romania, pp. 173–182. Theta Series in Advanced Mathematics, vol. 14. Theta, Bucharest, 2009) , Eells and Fuglede (Harmonic maps between Riemannian polyhedra. In: Cambridge Tracts in Mathematics, vol. 142. Cambridge University Press, Cambridge, 2001) and Sturm (Probability measures on metric spaces of nonpositive curvature. Probability measures on metric spaces of nonpositive curvature. In: Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), pp. 357–390. Contemporary Mathematics, vol. 338. American Mathematical Society, Providence, 2003). As an application, we give a Liouville’s theorem for harmonic maps described by Markov chains into \(2\)-uniformly convex space satisfying such a geometric condition. An alternative proof of the Jensen’s inequality over Banach spaces is also presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Afsari, B.: Riemannian \(L^p\) center of mass: existence, uniqueness, and convexity. Proc. Am. Math. Soc. 139(2), 655–673 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ball, K., Carlen, E.A., Lieb, E.H.: Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math. 115(3), 463–482 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)

  4. Clarkson, J.A.: Uniformly convex spaces. Trans. Am. Math. Soc. 40(3), 396–414 (1936)

    Article  MathSciNet  Google Scholar 

  5. Dor, L.E.: Potentials and isometric embeddings in \(L_{1}\). Israel J. Math. 24(3–4), 260–268 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  6. Eells, J. Jr., Fuglede, B.: Harmonic maps between Riemannian polyhedra. (English summary) With a preface by M. Gromov. In: Cambridge Tracts in Mathematics, vol. 142. Cambridge University Press, Cambridge (2001)

  7. Gromov, M.: Hyperbolic groups. In: Essays in Group Theory, pp. 75–263. Mathematical Sciences Research Institute Publications, vol. 8. Springer, New York (1987)

  8. Kendall, W.S.: Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence. In: Proceedings of London Mathematical Society (3), vol. 61, issue no. 2, pp. 371–406. (1990)

  9. Kendall, W.S.: From stochastic parallel transport to harmonic maps. In: New directions in Dirichlet forms, pp. 49–115. AMS/IP Studies in Advanced Mathematics, vol. 8. American Mathematical Society, Providence (1998)

  10. Korevaar, N.J., Schoen, R.M.: Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom. 1, 561–659 (1993)

    MATH  MathSciNet  Google Scholar 

  11. Kuwae, K.: Variational convergence for convex functionals (2013, preprint)

  12. Kuwae, K.: Jensen’s inequality over CAT\((\kappa )\)-space with small diameter. In: Proceedings of Potential Theory and Stochastics, Albac Romania, pp. 173–182. Theta Series in Advanced Mathematics, vol. 14. Theta, Bucharest (2009)

  13. Kuwae, K., Sturm, K.-Th.: On a Liouville type theorem for harmonic maps to convex spaces via Markov chains. In: Proceedings of German–Japanese symposium in Kyoto (2006), pp. 177–192. RIMS Kôkyûroku Bessatsu, vol. B6. (2008)

  14. Lim, T.-C., Xu, H.K., Xu, Z.B.: Some \(L^p\) inequalities and their applications to fixed point theory and approximation theory. In: Progress in Approximation Theory, pp. 609–624. Academic Press, Boston (1991)

  15. Naor, A., Silberman, L.: Poincaré inequalities, embeddings, and wild groups. Compos. Math. 147(5), 1546–1572 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ohta, S.- I.: Convexities of metric spaces. Geom. Dedicata 125(1), 225–250 (2007)

  17. Ohta, S.-I.: Extending Lipschitz and Hölder maps between metric spaces. Positivity 13(2), 407–425 (2009)

    Google Scholar 

  18. Ohta, S.-I.: Barycenters in Alexandrov spaces of curvature bounded below. Adv. Geom. 12(4) 571–587 (2012)

    Google Scholar 

  19. Prus, B., Smarzewski, R.: Strongly unique best approximations and centers in uniformly convex spaces. J. Math. Anal. Appl. 121(1), 10–21 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  20. Sturm, K-Th: Nonlinear Markov operators associated with symmetric Markov kernels and energy minimizing maps between singular spaces. Calc. Var. Partial Differ. Equ. 12(4), 317–357 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. Sturm, K-Th: Nonlinear Markov operators, discrete heat flow, and harmonic maps between singular spaces. Potential Anal. 16(4), 305–340 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Sturm, K-Th: Nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature. Ann. Probab. 30(3), 1195–1222 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Sturm, K.-Th.: Probability measures on metric spaces of nonpositive curvature. In: Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), pp. 357–390. Contemporary Mathematics, vol. 338. American Mathematical Society, Providence (2003)

  24. Thakur, B.S., Jung, J.S.: Fixed points of a certain class of mappings in uniformly convex Banach spaces. Bull. Korean Math. Soc. 34(3), 385–394 (1997)

    MATH  MathSciNet  Google Scholar 

  25. Villani, C.: Optimal transport, old and new. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338. Springer, Berlin (2009)

  26. Xu, H.-K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16(12), 1127–1138 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  27. Yang, L.: Riemannian median and its estimation, (English summary). LMS J. Comput. Math. 13, 461–479 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author thanks to Dr. Kei Funano for valuable comments to the draft of this paper and thanks also the anonymous referee for his/her suggestion to refer [1, 27], which yields some significant improvement of the draft.

Author information

Authors and Affiliations

  1. Department of Mathematics and Engineering, Graduate School of Science and Technology, Kumamoto University, Kumamoto, 860-8555, Japan

    Kazuhiro Kuwae

Authors
  1. Kazuhiro Kuwae
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kazuhiro Kuwae.

Additional information

Communicated by J. Jost.

The authors are partially supported by a Grant-in-Aid for Scientific Research No. 22340036 from the Japan Society for the Promotion of Science.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kuwae, K. Jensen’s inequality on convex spaces. Calc. Var. 49, 1359–1378 (2014). https://doi.org/10.1007/s00526-013-0625-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-013-0625-5

Mathematics Subject Classification (2000)

Search

Navigation