We establish the stability near a Euclidean ball of two conjectured inequalities: the dimensional Brunn-Minkowski inequality for radially symmetric log-concave measures in R n , and of the log-Brunn-Minkowski inequality.

Abstract. We discuss optimal constants in a recent result of Rudelson and Vershynin on marginal densities. We show that if f is a probability density on R n of the form f (x) = n i=1 f i (x i ), where each f i is a density on R, say bounded by one, then the density of any marginal π E (f ) is bounded by 2 k/2 , where k is the dimension of E. The proof relies on an adaptation of Ball's approach to cube slicing, carried out for functions. Motivated by inequalities for dual affine quermassintegrals, we also prove an isoperimetric inequality for certain averages of the marginals of such f for which the cube is the extremal case.

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We show the existence of a net near the sphere, such that the values of any matrix on the sphere and on the net are compared via a regularized Hilbert-Schmidt norm, which we introduce. This allows to construct an efficient net which controls the length of Ax for any random matrix A with independent columns (no other assumptions are required). As a consequence we show that the smallest singular value σ n (A) of an N × n random matrix A with i.i.d. mean zero, variance one entries enjoys the following small ball estimate, for any > 0:The proof of this result requires working with matrices whose rows are not independent, and, therefore, the fact that the theorem about discretization works for matrices with dependent rows, is crucial. Furthermore, in the case of the square n×n matrix A with independent entries having concentration function separated from 1, i.i.d. rows, and such that E||A|| 2 HS ≤ cn 2 , one hasfor any > 0. In addition, for > c √ n the assumption of i.i.d. rows is not required. Our estimates generalize the previous results of Rudelson and Vershynin [29], [30], which required the sub-gaussian mean zero variance one assumptions, as well as the work of Rebrova and Tikhomirov [25], where mean zero variance 1 and i.i.d. entries were required.

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