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The transportation method for proving concentration of measure results works directly for the cube. Here we find the best constant that can be found using this method which turns out to be better than those obtained by previous methods and which cannot be far from that which is best possible.
We prove the uniform concentration of Lebesgue measure phenomenon on the ball of ℓpn for 1 ≤ p ≤ 2. In particular, we give a first concentration inequality for Lebesgue measure on the ball of ℓ1n. An application is the lower exponential bound on the dimension of ℓ∞ admitting an isomorphic embedding of ℓ1 ...
We study the behaviour of constants in Khinchine-Kahane-type inequalities for polynomials in random vectors which have logarithmically concave distributions.
In this paper, we show how the methods from [B-G] may be adapted to establish Anderson localization for quasi-periodic lattice Schrödinger operators corresponding to the band model ℤ × {1, ..., b}. Recall that ‘Anderson localization’ means pure point spectrum with exponentially decaying eigenfunctions. We also discuss the issue of dynamical localization.
In this paper we study Euclidean projections of a p-convex body in ℝn. Precisely, we prove that for any integer k satisfying ln n ≤ k ≤ n/2, there exists a projection of rank k with the distance to the Euclidean ball not exceeding Cp(k/ln(1 + n/k))1/p−1/2, where Cp is an absolute positive constant depending only on p. Moreover, we...
Here we extend a result by J. Bourgain, J. Lindenstrauss, V.D. Milman on the number of random Minkowski symmetrizations needed to obtain an approximated ball, if we start from an arbitrary convex body in ℝn. We also show that the number of “deterministic” symmetrizations needed to approximate an Euclidean ball may be significantly smaller than the number of “random” ones.
We study the asymptotic behavior, as the dimension goes to infinity, of the volume of sections of the unit balls of the spaces ℓqn, 0 < q ≤ ∞. We compute the precise asymptotics of the average volume of central sections and then prove a concentration inequality of exponential type. For the case of non-central hyperplane sections of the cube,...
We prove a concentration inequality for functions, Lipschitz with respect to the Euclidean metric, on the ball of ℓpn, 1 ≤ p < 2 equipped with the normalized Lebesgue measure.
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