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Yet another note on the arithmetic-geometric mean inequality

Volume 253 / 2020

Zakhar Kabluchko, Joscha Prochno, Vladislav Vysotsky Studia Mathematica 253 (2020), 39-55 MSC: Primary 52A23, 60F05, 60F10; Secondary 46B06, 46B07. DOI: 10.4064/sm181014-16-3 Published online: 23 December 2019

Abstract

It was shown by E. Gluskin and V. D. Milman in [GAFA Lecture Notes in Math. 1807, 2003] that the classical arithmetic-geometric mean inequality can be reversed (up to a multiplicative constant) with high probability, when applied to coordinates of a point chosen with respect to the surface unit measure on a high-dimensional Euclidean sphere. We present two asymptotic refinements of this phenomenon in the more general setting of the surface probability measure on a high-dimensional $\ell _p$-sphere, and also show that sampling the point according to either the cone probability measure on $\ell _p$ or the uniform distribution on the ball enclosed by $\ell _p$ yields the same results. First, we prove a central limit theorem, which allows us to identify the precise constants in the reverse inequality. Second, we prove the large deviations counterpart to the central limit theorem, thereby describing the asymptotic behaviour beyond the Gaussian scale, and identify the rate function.

Authors

  • Zakhar KabluchkoInstitut für Mathematische Stochastik
    Westfälische Wilhelms-Universität Münster
    Münster, Germany
    e-mail
  • Joscha ProchnoInstitut für Mathematik
    & Wissenschaftliches Rechnen
    Karl-Franzens-Universität Graz
    Graz, Austria
    e-mail
  • Vladislav VysotskyDepartment of Mathematics
    University of Sussex
    Brighton, United Kingdom
    and
    St. Petersburg Department of Steklov Mathematical Institute
    St. Petersburg, Russia
    e-mail

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