Advertisement

On the Mean Width of Log-Concave Functions

  • Liran RotemEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2050)

Abstract

In this work we present a new, natural, definition for the mean width of log-concave functions. We show that the new definition coincides with a previous one by B. Klartag and V. Milman, and deduce some properties of the mean width, including an Urysohn type inequality. Finally, we prove a functional version of the finite volume ratio estimate and the low-\({M}^{{_\ast}}\) estimate.

Keywords

Mean Width Klartag Convex Bodies Quermassintegrals Urysohn Inequality 
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.

Notes

Acknowledgements

I would like to thank my advisor, Vitali Milman, for raising most of the questions in this paper, and helping me tremendously in finding the answers.

References

  1. 1.
    S. Artstein-Avidan, V. Milman, A characterization of the support map. Adv. Math. 223(1), 379–391 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    S. Artstein-Avidan, V. Milman, Hidden structures in the class of convex functions and a new duality transform. J. Eur. Math. Soc. (JEMS) 13(4), 975–1004 (2011)Google Scholar
  3. 3.
    S. Artstein-Avidan, B. Klartag, V. Milman, The santaló point of a function, and a functional form of the santaló inequality. Mathematika 51(1–2), 33–48 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    K. Ball, Isometric problems in l p and sections of convex sets. PhD thesis, Cambridge University (1986)Google Scholar
  5. 5.
    Y. Gordon, in On Milman’s Inequality and Random Subspaces which Escape Through a Mesh in \({\mathbb{R}}^{n}\), ed. by J. Lindenstrauss, V. Milman. Geometric Aspects of Functional Analysis, vol. 1317 (Springer, Berlin, 1988), pp. 84–106Google Scholar
  6. 6.
    B. Klartag, V. Milman, Geometry of log-concave functions and measures. Geometriae Dedicata 112(1), 169–182 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    R. McEliece, in The Theory of Information and Coding. Encyclopedia of Mathematics and Its Applications, vol. 86, 2nd edn. (Cambridge University Press, London, 2002)Google Scholar
  8. 8.
    V. Milman, Almost euclidean quotient spaces of subspaces of a Finite-Dimensional normed space. Proc. Am. Math. Soc. 94(3), 445–449 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    V. Milman, Geometrical inequalities and mixed volumes in the local theory of banach spaces. (Colloquium in honor of Laurent Schwartz, vol. 1 (Palaiseau, 1983). Astérisque 131, 373–400 (1985)Google Scholar
  10. 10.
    V. Milman, in Random Subspaces of Proportional Dimension of Finite Dimensional Normed Spaces: Approach Through the Isoperimetric Inequality, ed. by N. Kalton, E. Saab. Banach Spaces (Columbia, MO, 1984). Lecture Notes in Math., vol. 1166 (Springer, Berlin, 1985), pp. 106–115Google Scholar
  11. 11.
    V. Milman, in Geometrization of Probability, ed. by M. Kapranov, Y. Manin, P. Moree, S. Kolyada, L. Potyagailo. Geometry and Dynamics of Groups and Spaces, vol. 265 (Birkhäuser, Basel, 2008), pp. 647–667Google Scholar
  12. 12.
    A. Pajor, N. Tomczak-Jaegermann, Subspaces of small codimension of Finite-Dimensional banach spaces. Proc. Am. Math. Soc. 97(4), 637–642 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    G. Pisier, in The Volume of Convex Bodies and Banach Space Geometry. Cambridge Tracts in Mathematics, vol. 94 (Cambridge University Press, Cambridge, 1989)Google Scholar
  14. 14.
    R. Schneider, in Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and Its Applications, vol. 44 (Cambridge University Press, Cambridge, 1993)Google Scholar
  15. 15.
    S. Szarek, On kashin’s almost euclidean orthogonal decomposition of \({l}_{n}^{1}\). Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 26(8), 691–694 (1978)Google Scholar
  16. 16.
    S. Szarek, N. Tomczak-Jaegermann, On nearly euclidean decomposition for some classes of banach spaces. Compos. Math. 40(3), 367–385 (1980)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations