Journal of Mathematical Sciences

, Volume 238, Issue 4, pp 484–494 | Cite as

On Ƶp-Norms of Random Vectors

  • R. LatałaEmail author

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.


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  1. 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).Google Scholar
  2. 2.
    S. Artstein, V. D. Milman, and S. J. Szarek, “Duality of metric entropy,” Ann. Math., 159, 1313–1328 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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.Google Scholar
  4. 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).Google Scholar
  5. 5.
    P. Hitczenko, “Domination inequality for martingale transforms of a Rademacher sequence,” Israel J. Math., 84, 161–178 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    R. Latała, “Weak and strong moments of random vectors,” Marcinkiewicz Centenary Volume, Banach Center Publ., 95, 115–121 (2011).Google Scholar
  7. 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).Google Scholar
  8. 8.
    M. Ledoux and M. Talagrand, Probability in Banach Spaces. Isoperimetry and Processes, Springer, Berlin (1991).CrossRefzbMATHGoogle Scholar
  9. 9.
    E. Lutvak and G. Zhang, “Blaschke–Santaló inequalities,” J. Diff. Geom., 47, 1–16 (1997).CrossRefzbMATHGoogle Scholar
  10. 10.
    G. Paouris, “Concentration of mass on convex bodies,”Geom. Funct. Anal., 16, 1021–1049 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    V. N. Sudakov, “Gaussian measures, Cauchy measures and ε-entropy,” Soviet Math. Dokl., 10, 310–313 (1969).zbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of WarsawWarsawPoland

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