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1. CMB 2014 (vol 58 pp. 150)

Ostrovskii, Mikhail I.
 Connections Between Metric Characterizations of Superreflexivity and the Radon-NikodÃ½ Property for Dual Banach Spaces Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon-NikodÃ½m property (RNP) for dual spaces in terms of the infinite diamond. This paper is devoted to further study of relations between metric characterizations of superreflexivity and the RNP for dual spaces. The main result is that finite subsets of any set $M$ whose embeddability characterizes the RNP for dual spaces, characterize superreflexivity. It is also observed that the converse statement does not hold, and that $M=\ell_2$ is a counterexample. Keywords:Banach space, diamond graph, finite representability, metric characterization, Radon-NikodÃ½m property, superreflexivityCategories:46B85, 46B07, 46B22

2. CMB 2011 (vol 54 pp. 726)

Ostrovskii, M. I.
 Auerbach Bases and Minimal Volume Sufficient Enlargements Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a sufficient enlargement for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P\colon Y\to X$ such that $P(B_Y)\subset A$. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have exotic'' minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having exotic'' minimal-volume sufficient enlargements in terms of Auerbach bases. Keywords:Banach space, Auerbach basis, sufficient enlargementCategories:46B07, 52A21, 46B15

3. CMB 2007 (vol 50 pp. 138)

Sari, Bünyamin
 On the Structure of the Set of Symmetric Sequences in Orlicz Sequence Spaces We study the structure of the sets of symmetric sequences and spreading models of an Orlicz sequence space equipped with partial order with respect to domination of bases. In the cases that these sets are small'', some descriptions of the structure of these posets are obtained. Categories:46B20, 46B45, 46B07

4. 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 convexityCategories:46B07, 46B09, 46B20, 52A21
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