On Convex Bodies and Log-Concave Probability Measures with Unconditional Basis

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Empirical Processes with a Bounded Ψ1 Diameter (2010, background): “For the first part, note that by the Bobkov-Nazarov Theorem [7, 16], if N ∼ nα and if we denote the j−th coordinate of Xi by Xi(j), then with high probability, for every 1 ≤ i ≤ N and 1 ≤ j ≤ n, (Xi(j))∗ ≤ cα log(en/j).”
L_p moments of random vectors via majorizing measures (2005, background): “These moments were studied for a random vector uniformly distributed in an isotropic 1unconditional convex body in [2], and for a vector uniformly distributed in the unit ball of a Schatten trace class in [12], where it was proved that when s ≤ c√n, Ms is of the same order as M2 (up to constant not ”
A probabilistic approach to the geometry of the ℓᵨⁿ-ball (2005, background): “Milman was solved in very special cases such as zonoids (Paouris [23]) and unconditional bodies (Bobkov and Nazarov [11] show that the main diagonal is ψ2).”
Geometry of Log-concave Functions and Measures (2005, background): “As log-concave measures retain some features of uniform measures on convex bodies, many results on uniform measures on convex bodies may be generalized to log-concave measures (two samples among many are [Bo] and [ BN ]).”
Concentration of mass on convex bodies (2006, background): “Bobkov and Nazarov (see [7] and [8]) have clarified the picture of the volume distribution on isotropic unconditional convex bodies.”