Concentration of the Ratio between the Geometric and Arithmetic Means

This paper is motivated by the article [GluMi] by E. Gluskin and V. Milman, who considered the variant ∏n i=1 |yi| ≤ √ n−1 ∑n i=1 y 2 i of the AM-GM inequality in the equal weights case. Roughly speaking, they showed that the ratio ∏n i=1 |yi|/ √ n−1 ∑n i=1 y 2 i is bounded below by 0.394 asymptotically in n and with high probability, where probability refers to Haar measure on the euclidean unit sphere S 2 . Thus, the geometric and arithmetic means are comparable quantities on “large sets”, provided n is large. For the most part, we focus on the standard AM-GM inequality ∏n i=1 |x| αi,n i ≤ ∑n i=1 αi,n|x|i, where αi,n > 0 and ∑n i=1 αi,n = 1. We shall see that in some cases the concentration of measure phenomenon does take place: For certain sequences of weights, which include the