Skip to main content

Interpolations, Convexity and Geometric Inequalities

Part of the Lecture Notes in Mathematics book series (LNM,volume 2050)

Abstract

We survey some interplays between spectral estimates of Hörmander-type, degenerate Monge-Ampère equations and geometric inequalities related to log-concavity such as Brunn-Minkowski, Santaló or Busemann inequalities.

Keywords

  • Unit Ball
  • Convex Body
  • Plurisubharmonic Function
  • Complex Interpolation
  • Reinhardt Domain

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, B. Klartag, V.D. Milman, The santaló point of a function, and a functional form of the santaló inequality. Mathematika 51(1–2), 33–48 (2004)

    Google Scholar 

  2. K. Ball, Isometric problems in p and sections of convex sets. Ph.D. dissertation, Cambridge (1986)

    Google Scholar 

  3. K. Ball, Logarithmically concave functions and sections of convex sets in \({\mathbb{R}}^{n}\). Studia Math. 88(1), 69–84 (1988)

    Google Scholar 

  4. K. Ball, F. Barthe, A. Naor, Entropy jumps in the presence of a spectral gap. Duke Math. J. 119(1), 41–63 (2003)

    Google Scholar 

  5. B. Berndtsson, Prekopa’s theorem and kiselman’s minimum principle for plurisubharmonic functions. Math. Ann. 312(4), 785–792 (1998)

    Google Scholar 

  6. B. Berndtsson, Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains. Ann. Inst. Fourier (Grenoble) 56, 1633–1662 (2006)

    Google Scholar 

  7. B. Berndtsson, Curvature of vector bundles associated to holomorphic fibrations. Ann. Math. (2) 169(2), 531–560 (2009)

    Google Scholar 

  8. S.G. Bobkov, M. Ledoux, From Brunn-Minkowski to Brascamp-Lieb and to logarithmic sobolev inequalities. Geom. Funct. Anal. 10(5), 1028–1052 (2000)

    Google Scholar 

  9. H.J. Brascamp, E.H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22(4), 366–389 (1976)

    Google Scholar 

  10. H. Busemann, A theorem on convex bodies of the Brunn-Minkowski type. Proc. Nat. Acad. Sci. U.S.A. 35, 27–31 (1949)

    Google Scholar 

  11. A. Colesanti, From the Brunn-Minkowski inequality to a class of Poincaré-type inequalities. Comm. Contemp. Math. 10(5), 765–772 (2008)

    Google Scholar 

  12. D. Cordero-Erausquin, Santaló’s inequality on \({\mathbb{C}}^{n}\) by complex interpolation. C. R. Math. Acad. Sci. Paris 334, 767–772 (2002)

    Google Scholar 

  13. D. Cordero-Erausquin, On Berndtsson’s generalization of Prékopa’s theorem. Math. Z. 249(2), 401–410 (2005)

    Google Scholar 

  14. D. Cordero-Erausquin, M. Fradelizi, B. Maurey, The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems. J. Funct. Anal. 214(2), 410–427 (2004)

    Google Scholar 

  15. L. Hörmander, in Notions of Convexity. Progress in Mathematics, vol. 127 (Birkhäuser, Boston, 1994)

    Google Scholar 

  16. B. Klartag, in Marginals of Geometric Inequalities. Geometric Aspects of Functional Analysis, Lecture Notes in Math., vol. 1910 (Springer, Berlin, 2007), pp. 133–166

    Google Scholar 

  17. R. Rochberg, Interpolation of Banach spaces and negatively curved vector bundles. Pac. J. Math. 110(2), 355–376 (1984)

    Google Scholar 

  18. S. Semmes, Interpolation of Banach spaces, differential geometry and differential equations, Rev. Mat. Iberoamericana 4, 155–176 (1988)

    Google Scholar 

Download references

Acknowledgements

We thank Yanir Rubinstein and Bo Berndtsson for interesting, related discussions. Bo’az Klartag was supported in part by the Israel Science Foundation and by a Marie Curie Reintegration Grant from the Commission of the European Communities.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dario Cordero-Erausquin .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Cordero-Erausquin, D., Klartag, B. (2012). Interpolations, Convexity and Geometric Inequalities. 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_9

Download citation