Concentration phenomena in high dimensional geometry

The purpose of this note is to present several aspects of concentration phenomena in high dimensional geometry. At the heart of the study is a geometric analysis point of view coming from the theory of high dimensional convex bodies. The topic has a broad audience going from algorithmic convex geometry to random matrices. We have tried to emphasize different problems relating these areas of research. Another connected area is the study of probability in Banach spaces where some concentration phenomena are related with good comparisons between the weak and the strong moments of a random vector. 1 Convex geometry and log-concave measures A function f : R → R+ is said to be log-concave if ∀x, y ∈ R,∀θ ∈ [0, 1], f((1− θ)x+ θy) ≥ f(x)1−θf(y)θ Define a measure μ with a log-concave density f ∈ L 1 , the Prékopa-Leindler inequality [68, 59] implies that it satisfies: for every compact sets A,B ⊂ R, for every θ ∈ [0, 1], μ((1− θ)A+ θB) ≥ μ(A)1−θμ(B)θ. (1) A measure satisfying (1) is said to be log-concave. A complete characterization of logconcave measures is well known. It has been done during the sixties and seventies and it is related to the work of [46, 67, 68, 59, 15], see also the surveys [21, 61]. In [15] it is proved that a measure is log-concave if and only if it is absolutely continuous with respect to the Lebesgue measure on the affine subspace generated by its convex support, with logconcave locally integrable density. Classical examples are the case of product of exponential distributions, f(x) = 1 2n exp(−|x|1), the Gaussian measure, f(x) = 1 (2π)n/2 exp(−|x| 2 2/2), the uniform measure on a convex body, f(x) = 1K(x). Moreover, it is well known [22, 68, 59, 13] that the class of log-concave measures is stable under convolution and linear transformations. It is important to notice that the class of uniform distribution on a convex body is stable under linear transformation but not under convolution. This is one among several reasons why it is preferable to work with log-concave measures.