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The dimension of almost spherical sections of convex bodies

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References

  1. Barnette, D., The minimum number of vertices of a simple polytope.Israel J. Math., 10 (1971), 121–125.

    MATH  MathSciNet  Google Scholar 

  2. Bennett, G., Dor, L. E., Goodman, V., Johnson, W. B. &Newman, C. M., On uncomplemented subspaces ofL p , 1<p<2.Israel J. Math., 26 (1977), 178–187.

    Article  MathSciNet  MATH  Google Scholar 

  3. Davis, W. J., Dean, D. W. &Singer, I., Complemented subspaces and A systems in Banach spaces,Israel J. Math., 6 (1968), 303–309.

    MathSciNet  MATH  Google Scholar 

  4. Day, M. M.,Normed linear spaces, Springer Verlag 1973.

  5. Dinghas, A., Einfacher beweis der isoperimetrischen Eigenschaft der Kugel in Riemannschen Räumen konstanter Krümmung.Math. Nachr., 2 (1949), 148–162.

    MATH  MathSciNet  Google Scholar 

  6. Dvoretzky, A., Some results on convex bodies and Banach spaces.Proc. Int. Symp. on linear spaces, Jerusalem 1961, 123–160.

  7. Dvoretzky, A. &Rogers, C. A., Absolute and unconditional convergence in normed linear spaces.Proc. Nat. Acad. Sci. U.S.A., 36 (1950), 192–197.

    MathSciNet  MATH  Google Scholar 

  8. Enflo, P., Lindenstrauss, J. &Pisier, G., On the three space problem.Math. Scand., 36 (1975), 199–210.

    MathSciNet  MATH  Google Scholar 

  9. Figiel, T., A short proof of Dvoretzky's theorme on almost spherical sections.Compositio Math., 33 (1976), 297–301.

    MATH  MathSciNet  Google Scholar 

  10. Figiel, T., Lindenstrauss, J. &Milman V. D., The dimension of almost spherical sections of convex bodies.Bull. Amer. Math. Soc., 82 (1976), 575–578.

    Article  MathSciNet  MATH  Google Scholar 

  11. Figiel, T. &Pisier, G., Séries aléatoires dans les espaces uniformément convexes ou uniformément lisses.C. R. Acad. Sci. Paris, 279 (1974), 611–614.

    MathSciNet  MATH  Google Scholar 

  12. Garling, D. J. H. &Gordon, Y., Relations between some constants associated with finite-dimensional Banach spaces.Israel J. Math., 9 (1971), 346–361.

    MathSciNet  MATH  Google Scholar 

  13. Grothendieck, A., Resumé de la theorie metrique des produits tensoriels topologique.Bol. Soc. Matem. São Paulo, 8 (1956), 1–79.

    MATH  Google Scholar 

  14. Hadwiger, H.,Vorlesung über Inhalt, Oberfläche und Isoperimetric. Springer Verlag 1957.

  15. John, F. Extremum problems with inequalities as subsidiary conditions. Courant anniversary volume. Interscience, New York 1948, 187–204.

    Google Scholar 

  16. Johnson, W. B., A reflexive Banach space which is not sufficiently Euclidean.Studia Math., 55 (1976), 201–205.

    MATH  MathSciNet  Google Scholar 

  17. Kadec, M. I. &Snobar, M. G., Certain functionals on the Minkowsky compactum.Mat. Zametki, 10 (1971), 694–696 (translated from Russian).

    MathSciNet  Google Scholar 

  18. Kahane, J. P.,Some random series of functions. Health Mathematical Monographs 1968.

  19. Krivine, J. L., Sous-espaces de dimension fini des espaces de Banach reticulés.Ann. of Math., 104 (1976), 1–29.

    Article  MATH  MathSciNet  Google Scholar 

  20. Kwapien, S., Isomorphic characterization of inner product spaces by orthogonal series with vector valued coefficients.Studia Math., 44 (1972), 583–595.

    MATH  MathSciNet  Google Scholar 

  21. Larman, D. G. &Mani, P., Almost ellipsoidal sections and projections of convex bodies.Proc. Cambridge Philos. Soc. 77 (1975), 529–546.

    Article  MathSciNet  MATH  Google Scholar 

  22. Levy, P.,Problèmes concrets d'analyse functionelle, Gauthier Villars Paris 1951.

    Google Scholar 

  23. Lindenstrauss, J. &Pelczynski, A., Absolutely summing operators in £ p spaces and their applications.Studia Math., 29 (1968), 275–326.

    MathSciNet  MATH  Google Scholar 

  24. Lindenstrauss, J. &Tzafriri, L., On the complemented subspaces problem.Israel J. Math., 9 (1971), 263–269.

    MathSciNet  MATH  Google Scholar 

  25. Maleev, R. P. &Troyanski, S. L., On the moduli of convexity and smoothness of Orlicz spaces.Studia Math., 54 (1975), 131–141.

    MathSciNet  MATH  Google Scholar 

  26. Maurey, B., Un théorème de prolongement.C. R. Acad. Sci. Paris., Sér A-B, 279 (1974), 329–332.

    MATH  MathSciNet  Google Scholar 

  27. Maurey, B. &Pisier, G., Caractérisation d'une classe d'espaces de Banach par de propriétés des series aléatoires vectorielles.C. R. Acad. Sci. Paris, 277 (1973), 687–690.

    MathSciNet  MATH  Google Scholar 

  28. Maurey, B. &Pisier, G., Series de variables aléatoires vectorielles independantes et propriétés geometriques des espaces de Banach.Studia Math. 58 (1976), 45–90.

    MathSciNet  MATH  Google Scholar 

  29. Milman, V. D., A new proof of the theorem of A. Dvoretzky on sections of convex bodies.Funct. Anal. Appl. 5 (1971), 28–37 (translated from Russian).

    MATH  MathSciNet  Google Scholar 

  30. Pelczynski, A. &Rosenthal, H. P., Localization techniques inL p spaces.Studia Math., 52 (1975), 263–289.

    MathSciNet  MATH  Google Scholar 

  31. Rutovitz, D., Some parameters associated with finite-dimensional Banach spaces.J. London Math. Soc., 40 (1965), 241–255.

    MATH  MathSciNet  Google Scholar 

  32. Szankowski, A., On Dvoretzky's theorem on almost spherical sections of convex bodies.Israel J. Math., 17 (1974), 325–338.

    MATH  MathSciNet  Google Scholar 

  33. Schmidt, E., Die Brunn-Minkowski ungleichung.Math. Nachr., 1 (1948), 81–157.

    MATH  MathSciNet  Google Scholar 

  34. Tomczak-Jaegermann, E., The moduli of smoothness and convexity and the Rademacher averages of trace classesS p , 1≤p<∞.Studia Math., 50 (1974), 163–182.

    MATH  MathSciNet  Google Scholar 

  35. Tzafriri, L., On Banach spaces with unconditional bases.Israel J. Math., 17 (1974), 84–93.

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Polish Academy of Sciences, Gdansk, Poland

    T. Figiel

  2. Hebrew University, Jerusalem, Israel

    J. Lindenstrauss

  3. Tel Aviv University, Tel Aviv, Israel

    V. D. Milman

Authors
  1. T. Figiel
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  2. J. Lindenstrauss
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  3. V. D. Milman
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Additional information

Research supported in part by NSF Grant MPS 74-07509-A02.

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Figiel, T., Lindenstrauss, J. & Milman, V.D. The dimension of almost spherical sections of convex bodies. Acta Math. 139, 53–94 (1977). https://doi.org/10.1007/BF02392234

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  • DOI: https://doi.org/10.1007/BF02392234

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