University of Sussex
Browse
agm_inequalities4_formatted.pdf (442.21 kB)

Yet another note on the arithmetic-geometric mean inequality

Download (442.21 kB)
journal contribution
posted on 2023-06-09, 17:49 authored by Zakhar Kabluchko, Joscha Prochno, Vladislav VysotskiyVladislav Vysotskiy
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 here 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 behavior beyond the Gaussian scale, and identify the rate function.

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Studia Mathematica

ISSN

0039-3223

Publisher

Polskiej Akademii Nauk, Instytut Matematyczny

Volume

253

Page range

39-55

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2019-05-16

First Open Access (FOA) Date

2019-05-17

First Compliant Deposit (FCD) Date

2019-05-15

Usage metrics

    University of Sussex (Publications)

    Categories

    No categories selected

    Exports

    RefWorks
    BibTeX
    Ref. manager
    Endnote
    DataCite
    NLM
    DC