A+ CATEGORY SCIENTIFIC UNIT

# Publishing house / Journals and Serials / Studia Mathematica / Online First articles

## Studia Mathematica

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

## Yet another note on the arithmetic-geometric mean inequality

### Volume 253 / 2020

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
University of Sussex
Brighton, United Kingdom
and
St. Petersburg Department of Steklov Mathematical Institute
St. Petersburg, Russia
e-mail

## Search for IMPAN publications

Query phrase too short. Type at least 4 characters.