Skip to main content
SpringerLink
Log in

On Ƶp-Norms of Random Vectors

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

To any n-dimensional random vector X we may associate its Lp-centroid body Ƶp (X) and the corresponding norm. We formulate a conjecture concerning the bound on the Ƶp (X)-norm of X and show that it holds under some additional symmetry assumptions. We also relate our conjecture to estimates of covering numbers and Sudakov-type minoration bounds.

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. S. Artstein-Avidan, A. Giannopoulos, and V. D. Milman, Asymptotic Geometric Analysis. Part I, Mathematical Surveys and Monographs, 202, Amer. Math. Soc., Providence, Rhode Island (2015).

  2. S. Artstein, V. D. Milman, and S. J. Szarek, “Duality of metric entropy,” Ann. Math., 159, 1313–1328 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Bobkov and F. L. Nazarov, “On convex bodies and log-concave probability measures with unconditional basis,” in: Geometric Aspects of Functional Analysis, Lect. Notes Math., 1807 (2003), pp. 53–69.

  4. S. Brazitikos, A. Giannopoulos, P. Valettas, and B. H. Vritsiou, Geometry of Isotropic Convex Bodies, Mathematical Surveys and Monographs, 196, Amer. Math. Soc., Providence, Rhode Island (2014).

  5. P. Hitczenko, “Domination inequality for martingale transforms of a Rademacher sequence,” Israel J. Math., 84, 161–178 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Latała, “Weak and strong moments of random vectors,” Marcinkiewicz Centenary Volume, Banach Center Publ., 95, 115–121 (2011).

  7. R. Latała, “On some problems concerning log-concave random vectors,” Convexity and Concentration, IMA Vol. Math. Appl., 161, 525–539, Springer, New York (2017).

  8. M. Ledoux and M. Talagrand, Probability in Banach Spaces. Isoperimetry and Processes, Springer, Berlin (1991).

    Book  MATH  Google Scholar 

  9. E. Lutvak and G. Zhang, “Blaschke–Santaló inequalities,” J. Diff. Geom., 47, 1–16 (1997).

    Article  MATH  Google Scholar 

  10. G. Paouris, “Concentration of mass on convex bodies,”Geom. Funct. Anal., 16, 1021–1049 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  11. V. N. Sudakov, “Gaussian measures, Cauchy measures and ε-entropy,” Soviet Math. Dokl., 10, 310–313 (1969).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Warsaw, Warsaw, Poland

    R. Latała

Authors
  1. R. Latała
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Latała.

Additional information

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 457, 2017, pp. 211–225.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Latała, R. On Ƶp-Norms of Random Vectors. J Math Sci 238, 484–494 (2019). https://doi.org/10.1007/s10958-019-04252-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-019-04252-7

Search

Navigation