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A quantitative finite-dimensional Krivine theorem. (English) Zbl 0612.46021
From the authors’ abstract: Measure concentration arguments are applied to get a power-type estimate for the dimension of almost $$\ell_ p$$ subspaces of isomorphs of $$\ell^ n_ p$$ and for the length of almost- symmetric sequences under a nonlinear-type condition.
Reviewer: A.J.Ellis

##### MSC:
 46B25 Classical Banach spaces in the general theory
Full Text:
##### References:
 [1] N. Alon and D. Milman,Embedding of l k in finite dimensional Banach spaces, Isr. J. Math.45 (1983), 265–280. · Zbl 0546.46015 [2] D. Amir and V. D. Milman,Unconditional and symmetric sets in n-dimensional normed spaces, Isr. J. Math.37 (1980), 3–20. · Zbl 0445.46011 [3] T. Figiel, J. Lindenstrauss and V. D. Milman,The dimensions of almost spherical sections of convex bodies, Acta Math.139 (1977), 53–94. · Zbl 0375.52002 [4] M. Gromov and V. D. Milman,A topological application of the isoperimetric inequality, Am. J. Math.105 (1983), 843–854. · Zbl 0522.53039 [5] M. Gromov and V. D. Milman,Brunn theorem and a concentration of volume phenomena for symmetric convex bodies, Geometrical Aspects of Functional Analysis, Seminar Notes, Tel Aviv, 1983–4. [6] V. I. Gurari, M. I. Kadec and V. I. Macaev,On Banach-Mazur distance between certain Minkowski spaces, Bull. Acad. Polon. Sci.13 (1965), 719–722. [7] J. L. Krivine,Sous espaces de dimension finis des espaces de Banach reticulés, Ann. of Math.104 (1976), 1–29. · Zbl 0329.46008 [8] D. R. Lewis,Finite dimensional subspaces of L p, Studia Math.63 (1978), 207–212. · Zbl 0406.46023 [9] B. Maurey,Construction de suites symétriques, Comptes Rendus Acad. Sci. Paris.288 (1979), A679-A681. · Zbl 0398.46019 [10] V. D. Milman,A new proof of the theorem of A. Dvoretzky on sections of convex bodies, Funct. Anal. Appl.5 (1971), 28–37 (transl. from Russian). [11] V. D. Milman and G. Schechtman,Asymptotic Theory of Finite Dimensional Banach Spaces, Springer Lecture Notes, to appear. · Zbl 0911.52002 [12] G. Pisier,On the dimension of the l p n -subspaces of Banach spaces, for 1<2, Trans. Am. Math. Soc.276 (1983), 201–211. · Zbl 0509.46016 [13] G. Schechtman,Levy type inequality for a class of finite metric spaces, inMartingale Theory in Harmonic Analysis and Applications, Cleveland 1981, Springer Lecture Notes in Math. No. 939, 1982, pp. 211–215.
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