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Distortion and asymptotic structure. (English) Zbl 1039.46007
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland (ISBN 0-444-51305-1/hbk). 1333-1360 (2003).
The famous distortion problem asks whether it is true that for every equivalent norm \(p\) on \(l_2\) and every \(\varepsilon > 0\) there is an infinite-dimensional subspace \(X \subset l_2\) such that \(\sup \{ \frac{p(x)}{p(y)}: x,y \in S(X)\} < 1+ \varepsilon\), where \(S(X)\) is the unit sphere of \(X\) in the original norm. An analogous statement for \(l_1\) and \(c_0\) is true (James), but the distortion problem itself has a negative answer (Odell, Schlumprecht). The first part of the survey gives an exposition of the history of the distortion problem, its role in Banach space theory and a method of its solution with a sketch of the proof. This part contains also James’ result, Milman’s results on Lipschitz functions stabilization, the construction and properties of Tsirelson’s and Schlumprecht’s spaces. The second part deals with asymptotic structure, which plays the role of a bridge between finite- and infinite-dimensional structures in Banach space theory.
For the entire collection see [Zbl 1013.46001].

MSC:
46B03 Isomorphic theory (including renorming) of Banach spaces
46B20 Geometry and structure of normed linear spaces
46B25 Classical Banach spaces in the general theory
46-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functional analysis
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