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1. CMB Online first
Connections between metric characterizations of superreflexivity and the Radon-NikodÃ½m property for dual Banach spaces |
Connections between metric characterizations of superreflexivity and the Radon-NikodÃ½m 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, superreflexivity Categories:46B85, 46B07, 46B22 |
2. CMB 2011 (vol 54 pp. 726)
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 enlargement Categories:46B07, 52A21, 46B15 |
3. CMB 2007 (vol 50 pp. 138)
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)
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 |