We establish a sharp concentration of mass inequality for isotropic convex bodies: there exists an absolute constant c > 0 such that if K is an isotropic convex body in R n , thenfor every t 1, where LK denotes the isotropic constant.

Abstract. We establish a small ball probability inequality for isotropic logconcave probability measures: there exist absolute constants c 1 , c 2 > 0 such that if X is an isotropic log-concave random vector in R n with ψ 2 constant bounded by b and if A is a non-zero n × n matrix, then for every ε ∈ (0, c 1 ) and y ∈ R n ,where c 1 , c 2 > 0 are absolute constants.

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We present an approach that allows one to bound the largest and smallest singular values of an N ×n random matrix with iid rows, distributed according to a measure on R n that is supported in a relatively small ball and for which linear functionals are uniformly bounded in L p for some p > 8, in a quantitative (non-asymptotic) fashion. Among the outcomes of this approach are optimal estimates of 1 ± c √ n/N not only in the case of the above mentioned measure, but also when the measure is log-concave or when it is a product measure of iid random variables with "heavy tails".

Let µ be a probability measure on R n with a bounded density f . We prove that the marginals of f on most subspaces are well-bounded. For product measures, studied recently by Rudelson and Vershynin, our results show there is a trade-off between the strength of such bounds and the probability with which they hold. Our proof rests on new affinely-invariant extremal inequalities for certain averages of f on the Grassmannian and affine Grassmannian. These are motivated by Lutwak's dual affine quermassintegrals for convex sets. We show that key invariance properties of the latter, due to Grinberg, extend to families of functions. The inequalities we obtain can be viewed as functional analogues of results due to Busemann-Straus, Grinberg and Schneider. As an application, we show that without any additional assumptions on µ, any marginal π E (µ), or a small perturbation thereof, satisfies a nearly optimal small-ball probability.

International audienceWe obtain optimal inequalities for the volume of the polar of random sets, generated for instance by the convex hull of independent random vectors in Euclidean space. Extremizers are given by random vectors uniformly distributed in Euclidean balls. This provides a random extension of the Blaschke–Santaló inequality which, in turn, can be derived by the law of large numbers. The method involves shadow systems, their connection to Busemann type inequalities, and how they interact with functional rearrangement inequalities