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Quotients of finite-dimensional Banach spaces; random phenomena. (English) Zbl 1057.46010
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland (ISBN 0-444-51305-1/hbk). 1201-1246 (2003).
This survey is devoted to properties of “random” subspaces/quotients of finite-dimensional normed spaces and to their applications.
The authors concentrate on the study of “random” phenomena of non-Euclidean nature. Since Gluskin proved a theorem about the diameter of the Minkowski compactum in 1981, “random” subspaces/quotients have played an important role in the asymptotic theory of finite dimensional normed spaces. They have been used to construct many examples of convex bodies with extremal behavior. In many cases, it has been shown that quantitative estimates, trivially obtained by comparison with the Euclidean case, are unimprovable.
Section 1 is an introduction, which contains a motivation and describes the construction of the paper. Section 2 contains preliminary results and a description of Gaussian quotients. In Section 3, the complete proof of Gluskin’s theorem is given. In Sections 4–6, the authors continue to investigate properties of “random” quotients of \(\ell_1^N\). They present many results including Gluskin’s and Szarek’s examples of spaces with large basis constants, Szarek’s example of a space without the well-bounded Dvoretzky-Rogers factorization, and results on the Gordon-Lewis property. As a powerful tool, used in many proofs, mixing operators are introduced and intensively studied in those sections.
Sections 7 and 8 deal with structural properties of finite- and infinite-dimensional Banach spaces. In Section 7, the authors adapt the methods used for quotients of \(\ell_1^N\) to the general case of a convex body (in a special position). The authors obtain quantitative estimates using corresponding volumetric parameters of a body. In Section 8, several results including the solution of the finite-dimensional homogeneous space problem by Bourgain and Mankiewicz-Tomczak-Jaegermann are presented. In Sections 9 and 10, the authors show how to use finite-dimensional constructions in the infinite-dimensional case. In particular, they prove that there exists a separable Banach space without Schauder basis and, also, provide an unexpected characterization of Hilbert spaces connected to the existence of Schauder bases. Finally, in Section 11, recent developments of the theory are briefly presented.
For the entire collection see [Zbl 1013.46001].

46B07 Local theory of Banach spaces
46-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functional analysis
46B08 Ultraproduct techniques in Banach space theory
46B09 Probabilistic methods in Banach space theory
46B20 Geometry and structure of normed linear spaces
46C15 Characterizations of Hilbert spaces
52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)