Yet another note on the arithmetic-geometric mean inequality

TL;DR

This paper extends the understanding of the reverse arithmetic-geometric mean inequality in high dimensions to various probability measures on $\, ext{l}_p$-spheres, providing precise constants and asymptotic behaviors through CLT and large deviations.

Contribution

It introduces two asymptotic refinements of the reverse inequality for different measures on high-dimensional $\, ext{l}_p$-spheres, including a CLT and large deviations analysis.

Findings

Established a central limit theorem for the reverse inequality constants.

Derived large deviations principles and identified the rate function.

Showed equivalence of results across different probability measures.

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 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.