Skip to main content
SpringerLink
Log in

Main Examples

  • Chapter
  • First Online:
Approaching the Kannan-Lovász-Simonovits and Variance Conjectures

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2131))

  • 884 Accesses

Abstract

In this second chapter the main examples known to satisfy the KLS conjecture, the square negative correlation property or the variance conjecture are provided. We also show Klartag’s results on unconditional convex bodies, which show that, up to a logarithmic factor, they verify the KLS conjecture and they verify the variance conjecture.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • 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

Institutional subscriptions

References

  1. D. Alonso-Gutiérrez, J. Bastero, The variance conjecture on some polytopes, in Asymptotic Geometric Analysis, Proceedings of the Fall 2010, Fields Institute Thematic Program, Fieds Institute Communications, vol. 68 (Springer, Berlin, 2013), pp. 1–20

    Google Scholar 

  2. M. Anttila, K. Ball, I. Perissinaki, The central limit problem for convex bodies. Trans. Am. Math. Soc. 355(12), 4723–4735 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. K. Ball, Isometric Problems in ℓ p and Sectios odf Convex Sets. Ph.D. thesis, Cambridge University, 1986

    Google Scholar 

  4. F. Barthe, D. Cordero-Erausquin, Invariances in variance estimates. Proc. Lond. Math. Soc. (3) 106(1), 33–64 (2013)

    Google Scholar 

  5. F. Barthe, P. Wolff, Remarks on non-interacting conservative spin systems: the case of gamma distributions. Stochast. Process. Appl. 119, 2711–2723 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. F. Barthe, O. Guédon, S. Mendelson, A. Naor, A probabilistic approach to the geometry of the ℓ p n-ball. Ann. Probab. 33(2), 480–513 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. S. Bobkov, Spectral gap and concentration for some spherically symmetric probability measures, in Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics, vol. 1807 (Springer, Berlin, 2003), pp. 37–43

    Google Scholar 

  8. C. Borell, Convex measures on locally convex spaces, Ark. Math. 12, 239–252 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  9. D. Hensley, Slicing convex bodies, bounds of slice area in terms of the body’s covariance. Proc. Am. Math. Soc. 79, 619–625 (1980)

    MATH  MathSciNet  Google Scholar 

  10. N. Huet, Spectral gap for some invariant log-concave probability measures. Mathematika 57(1), 51–62 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. R. Kannan, L. Lovász, M. Simonovits, Isoperimetric problems for convex bodies and a localization lemma. Discret. Comput. Geom. 13(3–4), 541–559 (1995)

    Article  MATH  Google Scholar 

  12. B. Klartag, A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Relat. Fields 145(1–2), 1–33 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. B. Klartag, Poincaré inequalities and moment maps. Ann. Fac. Sci. Toulouse Math. 22(1), 1–41 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  14. R. Latała, J.O. Wojtaszczyk, On the infimum convolution inequality. Stud. Math. 189(2), 147–187 (2008)

    Article  MATH  Google Scholar 

  15. E. Milman, On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177(1), 1–43 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. V.D. Milman, A. Pajor, in Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, GAFA Seminar 87–89, Springer Lecture Notes in Mathematics, vol. 1376 (1989), pp. 64–104

    Google Scholar 

  17. C.A. Rogers, G.C. Shephard, Convex bodies associated with a given convex body. J. Lond. Math. Soc. 33, 270–281 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  18. G. Schechtman, J. Zinn, Concentration on the ℓ p n ball, in Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 1745 (Springer, Berlin, 2000), pp. 245–256

    Google Scholar 

  19. S. Sodin, An isoperimetric inequality on the ℓ p balls. Ann. Inst. H. Poincar é Probab. Stat. 44(2), 362–373 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. J.O. Wojtaszczyk, The square negative correlation property for generalized Orlicz balls, in Geometric Aspects of Functional Analysis. Israel seminar, Lecture Notes in Mathematics, vol. 1910 (Springer, Berlin, 2007), pp. 305–313

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departament de Matemàtiques and IMAC, Universitat Jaume I, Castelló de la Plana, Spain

    David Alonso-Gutiérrez

  2. Departamento de Matemáticas and IUMA, Universidad de Zaragoza, Zaragoza, Spain

    Jesús Bastero

Authors
  1. David Alonso-Gutiérrez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jesús Bastero
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Alonso-Gutiérrez, D., Bastero, J. (2015). Main Examples. In: Approaching the Kannan-Lovász-Simonovits and Variance Conjectures. Lecture Notes in Mathematics, vol 2131. Springer, Cham. https://doi.org/10.1007/978-3-319-13263-1_2

Download citation

Publish with us

Policies and ethics

Search

Navigation