On the Equivalence Between Geometric and Arithmetic Means for Log-Concave Measures

Summary

Introduction. Let X be a random vector with log-concave distribution (for precise definitions see below). It is known that for any measurable seminorm and p, q > 0 the inequality holds with constants C_p,q depending only on p and q (see [4], Appendix III). In this paper we show that the above constants can be made independent of q, which is equivalent to the inequality

where IIXll_o is the geometric mean of IIXII. In the particular case in which X is uniformly distributed on some convex compact set in Rⁿ and the seminorm is given by some functional, inequality (1) was established by V. D. Milman and A. Pajor [3]. As a consequence of (1) we prove the result of Ullrich [6] concerning the equivalence of means for sums of independent Steinhaus random variables with vector coefficients, even though these random-variables are not log-concave (Corollary 2). To prove (1) we derive some estimates of log-concave measures of small balls (Corollary 1), which are of independent interest In the case of Gaussian random variables they were formulated and established in a weaker version in [5] and completelely proved in [2].

We say that a random vector X with values in E is log-concave if the distribution of X is log-concave. For a random vector X and a measurable seminorm ||.|| on E (i.e. Borel measurable, nonnegative, subadditive and positively homogeneous function on E) we define.

Let us begin with the following Lemma from [1].