Ψ α -ESTIMATES FOR MARGINALS OF LOG-CONCAVE PROBABILITY MEASURES A. GIANNOPOULOS, G. PAOURIS, AND P. VALETTAS Abstract. We show that a random marginal π F (μ) of an isotropic log-concave probability measure μ on R n exhibits better ψα-behavior: (i) If k ≤ √ n, then for a random F ∈ G n,k we have that π F (μ) is a ψ 2 - measure. We complement this result by showing that a random π F (μ) is, at the same time, supergaussian. (ii) If k = n δ , 1 2 <δ< 1, then for a random F ∈ G n,k we have that π F (μ) is a ψ α(δ) -measure, where α(δ)= 2δ 3δ-1 . 1. Introduction The purpose of this note is to provide estimates on the ψ α -behavior of ran- dom marginals of log-concave probability measures. We show that random k– dimensional projections of a high-dimensional measure of the log-concave class have better tail properties than the original measure. We give precise quantitative es- timates for every 1 ≤ k<n. A typical k-dimensional marginal is ψ 2 as long as k ≤ √ n; after this critical value we still have non-trivial information (α is always greater than a simple function of log n log k ) in full generality. This observation may be viewed as a continuation of the ideas and the tools that were developed in [17]. It is also parallel to the philosophy behind Klartag’s proof of the central limit theorem for convex bodies in [7] and [8] (see also [5] and [4]). A main ingredient in these works is the fact that appropriate marginals of log-concave measures in power-type dimensions (k ≃ n ǫ for some ǫ> 0) are approximately spherically-symmetric. As Klartag proves in [9] this phenomenon appears for a much wider class of probability measures and constitutes the measure analogue of Dvoretzky’s theorem on approxi- mately Euclidean sections of high-dimensional convex bodies. Actually, Dvoretzky’s theorem plays a crucial role in all these works, as well as in the present note. Recall that a probability measure µ on R n is called log-concave if for any Borel sets A,B in R n and any λ ∈ (0, 1), (1.1) µ(λA + (1 − λ)B) ≥ µ(A) λ µ(B) 1−λ . It is known (see [2]) that if µ is log-concave and if µ(H) < 1 for every hyperplane H, then µ has a density f = f µ , with respect to the Lebesgue measure, which is log-concave: log f is concave on its support {f> 0}. Date : July 20, 2010. 1991 Mathematics Subject Classification. Primary 52A20; Secondary 46B07. Key words and phrases. Log-concave probability measures, random marginals, isotropic constant. The second named author is partially supported by an NSF grant. 1