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Search: MSC category 46B09 ( Probabilistic methods in Banach space theory [See also 60Bxx] )

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1. CMB 2012 (vol 57 pp. 3)

Adamczak, Radosław; Latała, Rafał; Litvak, Alexander E.; Oleszkiewicz, Krzysztof; Pajor, Alain; Tomczak-Jaegermann, Nicole
A Short Proof of Paouris' Inequality
We give a short proof of a result of G.~Paouris on the tail behaviour of the Euclidean norm $|X|$ of an isotropic log-concave random vector $X\in\mathbb{R}^n,$ stating that for every $t\geq 1$, \[\mathbb{P} \big( |X|\geq ct\sqrt n\big)\leq \exp(-t\sqrt n).\] More precisely we show that for any log-concave random vector $X$ and any $p\geq 1$, \[(\mathbb{E}|X|^p)^{1/p}\sim \mathbb{E} |X|+\sup_{z\in S^{n-1}}(\mathbb{E} |\langle z,X\rangle|^p)^{1/p}.\]

Keywords:log-concave random vectors, deviation inequalities
Categories:46B06, 46B09, 52A23

2. CMB 2006 (vol 49 pp. 313)

Wagner, Roy
On the Relation Between the Gaussian Orthogonal Ensemble and Reflections, or a Self-Adjoint Version of the Marcus--Pisier Inequality
We prove a self-adjoint analogue of the Marcus--Pisier inequality, comparing the expected value of convex functionals on randomreflection matrices and on elements of the Gaussian orthogonal (or unitary) ensemble.

Categories:15A52, 46B09, 46L54

3. CMB 2003 (vol 46 pp. 242)

Litvak, A. E.; Milman, V. D.
Euclidean Sections of Direct Sums of Normed Spaces
We study the dimension of ``random'' Euclidean sections of direct sums of normed spaces. We compare the obtained results with results from \cite{LMS}, to show that for the direct sums the standard randomness with respect to the Haar measure on Grassmanian coincides with a much ``weaker'' randomness of ``diagonal'' subspaces (Corollary~\ref{sle} and explanation after). We also add some relative information on ``phase transition''.

Keywords:Dvoretzky theorem, ``random'' Euclidean section, phase transition in asymptotic convexity
Categories:46B07, 46B09, 46B20, 52A21

4. CMB 2000 (vol 43 pp. 368)

Litvak, A. E.
Kahane-Khinchin's Inequality for Quasi-Norms
We extend the recent results of R.~Lata{\l}a and O.~Gu\'edon about equivalence of $L_q$-norms of logconcave random variables (Kahane-Khinchin's inequality) to the quasi-convex case. We construct examples of quasi-convex bodies $K_n \subset \R$ which demonstrate that this equivalence fails for uniformly distributed vector on $K_n$ (recall that the uniformly distributed vector on a convex body is logconcave). Our examples also show the lack of the exponential decay of the ``tail" volume (for convex bodies such decay was proved by M.~Gromov and V.~Milman).

Categories:46B09, 52A30, 60B11

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