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Thin-shell concentration for convex measures

100%

Fradelizi M. , Guédon O. , Pajor A.

Studia Mathematica

2014

tom 223

nr 2

123-148

We prove that for s < 0, s-concave measures on ℝⁿ exhibit thin-shell concentration similar to the log-concave case. This leads to a Berry-Esseen type estimate for most of their one-dimensional marginal distributions. We also establish sharp reverse Hölder inequalities for s-concave measures.

Chevet type inequality and norms of submatrices

88%

Adamczak R. , Latała R. , Litvak A. , Pajor A. , Tomczak-Jaegermann N.

Studia Mathematica

2012

tom 210

nr 1

35-56

We prove a Chevet type inequality which gives an upper bound for the norm of an isotropic log-concave unconditional random matrix in terms of the expectation of the supremum of "symmetric exponential" processes, compared to the Gaussian ones in the Chevet inequality. This is used to give a sharp upper estimate for a quantity $Γ_{k,m}$ that controls uniformly the Euclidean operator norm of the submatrices with k rows and m columns of an isotropic log-concave unconditional random matrix. We apply these estimates to give a sharp bound for the restricted isometry constant of a random matrix with independent log-concave unconditional rows. We also show that our Chevet type inequality does not extend to general isotropic log-concave random matrices.