Yet another note on the arithmetic-geometric mean inequality
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.