Skip to main content
SpringerLink
Log in

Bourgain’s slicing problem and KLS isoperimetry up to polylog

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

We prove that Bourgain’s hyperplane conjecture and the Kannan-Lovász-Simonovits (KLS) isoperimetric conjecture hold true up to a factor that is polylogarithmic in the dimension.

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

  2. Bakry, D., Gentil, I., Ledoux, M., Analysis and geometry of Markov diffusion operators. Springer, 2014.

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

  4. Bauerschmidt, R., Bodineau, T., Log-Sobolev inequality for the continuum sine-Gordon model. Comm. Pure Appl. Math., 74(10) (2021), 2064–2113

  5. Barthe, F., Klartag, B., Spectral gaps, symmetries and log-concave perturbations. Bull. Hellenic Math. Soc., 64 (2020), 1–31

  6. Bobkov, S. G., Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab., Vol. 27 (1999), 1903–1921.

    Article  MathSciNet  Google Scholar 

  7. Bobkov, S. G., Koldobsky, A., On the central limit property of convex bodies. Geom. Aspects of Funct. Anal., Israel seminar (2001–02), Lecture Notes in Math., Vol. 1807, Springer, (2003), 44–52.

  8. Bourgain, J., On high-dimensional maximal functions associated to convex bodies. Amer. J. Math., vol. 108, (1986), 1467–1476.

    Article  MathSciNet  Google Scholar 

  9. Bourgain, J., Geometry of Banach spaces and harmonic analysis. Proceedings of the International Congress of Mathematicians, (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, (1987), 871–878.

  10. Bourgain, J., On the distribution of polynomials on high dimensional convex sets. Geom. Aspects of Funct. Anal., Israel seminar (1989–90), Lecture Notes in Math., Vol. 1469, Springer, (1991), 127–137.

  11. Bourgain, J., On the isotropy-constant problem for “Psi-\(2\)” bodies. Geom. Aspects of Funct. Anal., Israel seminar (2001–02), Lecture Notes in Math., Vol. 1807, Springer (2002), 114–121.

  12. Brazitikos, S., Giannopoulos, A., Valettas, P., Vritsiou, B. -H., Geometry of isotropic convex bodies. American Mathematical Society, 2014.

  13. Buser, P., A note on the isoperimetric constant. Ann. Sci. École Norm. Sup., Vol. 15, (1982), 213–230.

    Article  MathSciNet  Google Scholar 

  14. Caffarelli, L. A., Monotonicity properties of optimal transportation and the FKG and related inequalities. Comm. Math. Phys., 214(3) (2000), 547–563

  15. Cheeger, J., A lower bound for the smallest eigenvalue of the Laplacian. Problems in analysis, Princeton Univ. Press, (1970), 195–199.

  16. Chen, Y., Chewi, S., Salim, A., Wibisono, A., Improved analysis for a proximal algorithm for sampling. Proceedings of Thirty Fifth Conference on Learning Theory (COLT 2022), PMLR 178:2984-3014, 2022. arXiv:2202.06386

  17. Chen, Y., An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture. Geom. Funct. Anal. (GAFA), Vol. 31, (2021), 34–61.

    Article  MathSciNet  Google Scholar 

  18. Cordero-Erausquin, D.,Fradelizi, M., Maurey, B., The\((B)\)conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems. J. Funct. Anal. Vol. 214, (2004), no. 2, 410–427.

    Article  MathSciNet  Google Scholar 

  19. Davidovič, Ju. S., Korenbljum, B. I., Hacet, B. I., A certain property of logarithmically concave functions. Dokl. Akad. Nauk SSSR, 185 (1969), 1215-1218. English translation in Soviet Math. Dokl., 10 (1969), 477-480

  20. Davies, E. B., Spectral theory and differential operators. Cambridge Studies in Advanced Mathematics, Vol. 42, Cambridge University Press, 1995.

  21. Diaconis, P., Freedman, D., Asymptotics of graphical projection pursuit. Ann. Statist., 12, (1984), 793–815.

    Article  MathSciNet  Google Scholar 

  22. Eidelman, Y., Milman, V., Tsolomitis, A., Functional analysis. An introduction. Graduate Studies in Mathematics, 66. American Mathematical Society, 2004.

  23. Eldan, R., Thin shell implies spectral gap via a stochastic localization scheme. Geom. Funct. Anal. (GAFA), Vol. 23, (2013), 532–569.

    Article  MathSciNet  Google Scholar 

  24. Eldan, R., Distribution of Mass in Convex Bodies. Ph.D. thesis, Tel Aviv University, 2012. Available at http://www.math.tau.ac.il/images/dissertations/eldan_ronen.pdf

  25. Eldan, R., Klartag, B., Approximately gaussian marginals and the hyperplane conjecture. Proc. of a workshop on “Concentration, Functional Inequalities and Isoperimetry”, Contermporary Mathematics 545, Amer. Math. Soc., (2011), 55–68.

  26. Eldan, R., Lehec, J., Bounding the norm of a log-concave random vector via thin-shell estimates. Geom. Aspects of Funct. Anal., Israel seminar (2011–13), Lecture Notes in Math., 2116, 107–122.

  27. Fleury, B., Concentration in a thin euclidean shell for log-concave measures. J. Func. Anal., Vol. 259, (2010), 832–841.

    Article  MathSciNet  Google Scholar 

  28. Guédon, O., Milman, E., Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures. Geom. Funct. Anal. (GAFA), Vol. 21, (2011), 1043–1068.

    Article  MathSciNet  Google Scholar 

  29. Jiang, H., Lee, Y.T., Vempala, S.S., A generalized central limit conjecture for convex bodies. Geom. Aspects of Funct. Anal., Israel seminar (2017–19), Lecture Notes in Math., Vol. 2266, Springer (2020), 1–41.

  30. Kannan, R., Lovász, L., Simonovits M., Isoperimetric problems for convex bodies and a localization lemma. Discr. Comput. Geom., Vol. 13, (1995), 541–559.

    Article  MathSciNet  Google Scholar 

  31. Klartag, B., On convex perturbations with a bounded isotropic constant. Geom. and Funct. Anal. (GAFA), Vol. 16, (2006), 1274–1290.

    Article  MathSciNet  Google Scholar 

  32. Klartag, B., A central limit theorem for convex sets. Invent. Math., Vol. 168, (2007), 91–131.

    Article  MathSciNet  Google Scholar 

  33. Klartag, B., Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal., Vol. 245, (2007), 284–310.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. Klartag, B., On Yuansi Chen’s work on the KLS conjecture. Lecture notes from a winter school at the Hausdorff Institute, January 2021. https://www.him.uni-bonn.de/fileadmin/him/Lecture_Notes/chen_lecture_notes.pdf

  36. Klartag, B., Milman, V., The slicing problem by Bourgain. To appear in Analysis at Large, a collection of articles in memory of Jean Bourgain, edited by A. Avila, M. Rassias and Y. Sinai, Springer, 2022.

  37. Klartag, B., Putterman, E., Spectral monotonicity under Gaussian convolution. To appear in Ann. Fac. Sci. Toulouse Math. arXiv:2107.09496

  38. Ledoux, M., Spectral gap, logarithmic Sobolev constant, and geometric bounds. Surveys in differential geometry. Vol. IX, Int. Press, (2004), 219–240.

  39. Lee, Y. T., Vempala, S., Eldan’s Stochastic Localization and the KLS Conjecture: Isoperimetry, Concentration and Mixing. 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2017), IEEE Computer Soc., (2017), 998–1007.

  40. Lee, Y. T., Vempala, S., Stochastic localization + Stieltjes barrier = tight bound for log-Sobolev. 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2018), ACM, (2018), 1122–1129.

  41. Lee, Y. T., Shen, R., Tian, K., Structured Logconcave Sampling with a Restricted Gaussian Oracle. Proceedings of Thirty Fourth Conference on Learning Theory (COLT 2021), PMLR 134:2993-3050, 2021. arXiv:2010.03106

  42. Paouris, G., Concentration of mass on convex bodies. Geom. Funct. Anal. (GAFA), Vol. 16, (2006), 1021–1049.

    Article  MathSciNet  Google Scholar 

  43. Sudakov, V. N., Typical distributions of linear functionals in finite-dimensional spaces of high-dimension. (Russian) Dokl. Akad. Nauk. SSSR, 243(6) (1978), 1402-1405. English translation in Soviet Math. Dokl., 19 (1978), 1578-1582.

  44. Villani, C., Topics in optimal transportation. American Mathematical Society, 2003.

Download references

Acknowledgments

The first-named author would like to thank Ronen Eldan and Eli Putterman for interesting discussions, and was partially supported by a grant from the Israel Science Foundation (ISF).

Author information

Authors and Affiliations

  1. Department of Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel

    Bo’az Klartag

  2. CEREMADE (UMR CNRS 7534), Université Paris-Dauphine, 75016, Paris, France

    Joseph Lehec

Authors
  1. Bo’az Klartag
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joseph Lehec
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bo’az Klartag.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Klartag, B., Lehec, J. Bourgain’s slicing problem and KLS isoperimetry up to polylog. Geom. Funct. Anal. 32, 1134–1159 (2022). https://doi.org/10.1007/s00039-022-00612-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-022-00612-9

Search

Navigation