Israel Journal of Mathematics

, Volume 45, Issue 4, pp 265–280

Embedding oflk in finite dimensional Banach spaces

  • N. Alon
  • V. D. Milman
Article
  • 117 Downloads

Abstract

Letx1,x2, ...,xn ben unit vectors in a normed spaceX and defineMn=Ave{‖Σi=1nε1xi‖:ε1=±1}. We prove that there exists a setA⊂{1, ...,n} of cardinality\(\left| A \right| \geqq \left[ {\sqrt n /\left( {2^7 M_n } \right)} \right]\) such that {xi}i∈A is 16Mn-isomorphic to the natural basis oflk. This result implies a significant improvement of the known results concerning embedding oflk in finite dimensional Banach spaces. We also prove that for every ∈>0 there exists a constantC(∈) such that every normed spaceXn of dimensionn either contains a (1+∈)-isomorphic copy ofl2m for somem satisfying ln lnm≧1/2 ln lnn or contains a (1+∈)-isomorphic copy oflk for somek satisfying ln lnk>1/2 ln lnnC(∈). These results follow from some combinatorial properties of vectors with ±1 entries.

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References

  1. 1.
    N. Alon,On the density of sets of vectors, Discrete Math., to appear.Google Scholar
  2. 2.
    D. Amir and V. D. Milman,Unconditional and symmetric sets in n-dimensional normed spaces, Isr. J. Math.37 (1980), 3–20.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    G. Bennett, L. E. Dor, V. Goodman, W. B. Johnson and G. M. Newman,On uncomplemented subspaces of L p,1<p<2, Isr. J. Math.26 (1977), 178–187.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    B. Bollobás,Extremal Graph Theory, Academic Press, London and New York, 1978, pp. 52–53.MATHGoogle Scholar
  5. 5.
    J. Bourgain,On complemented l 1 sequences, preprint.Google Scholar
  6. 6.
    P. Erdös and J. Spencer,Probabilistic Methods in Combinatorics, Academic Press, New York and London, 1974, p. 18.MATHGoogle Scholar
  7. 7.
    P. Erdös, Chao Ko and R. Rado,Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Sec.,12 (48) (1961), 313–320.MATHCrossRefGoogle Scholar
  8. 8.
    T. Figiel and N. Tomczak-Jaegermann,Projections onto Hilbertian subspaces of Banach spaces, Isr. J. Math.33 (1979), 155–171.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    T. Figiel, J. Lindenstrauss and V. Milman,The dimension of almost spherical sections of convex bodies, Acta Math.139 (1977), 53–94.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    D. P. Giesy,On a convexity condition in normed linear spaces, Trans. Am. Math. Soc.125 (1966), 114–146.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    W. B. Johnson and G. Schechtman,On subspaces of L 1 with maximal distance to Euclidean space, inProc. Research Workshop on Banach Space Theory (Bor Luh Lin, ed.), University of Iowa, 1981, pp. 83–96.Google Scholar
  12. 12.
    W. B. Johnson and G. Schechtman,Embedding l pm into l 1n, Acta Math.149 (1982), 71–85.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    M. G. Karpovsky and V. Milman,Coordinate density of sets of vectors, Discrete Math.24 (1978), 177–184.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    D. R. Lewis,Finite dimensional subspaces of L p, Studia Math.63 (1978), 207–212.MATHMathSciNetGoogle Scholar
  15. 15.
    B. Maurey and G. Pisier,Caractérisation d’une classe d’espaces de Banach par des propriétés de séries aléatoires vectorièlles, C.R. Acad. Sci. Paris, Sér. A,277 (1973), 687–690.MATHMathSciNetGoogle Scholar
  16. 16.
    B. Maurey and G. Pisier,Séries de variables aléatoires vectorièlles indépendantes et propriétés géométriques des espaces de Banach, Studia Math.58 (1976), 45–90.MATHMathSciNetGoogle Scholar
  17. 17.
    V. Milman,Some remarks about embedding of l 1k in finite dimensional spaces, Isr. J. Math.43 (1982), 129–138.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    G. Pisier,On the dimension of the l pn-subspaces of Banach spaces, for 1≦p<2, preprint.Google Scholar
  19. 19.
    N. Sauer,On the density of families of sets, J. Comb. Theory, Ser. A,13 (1972), 145–147.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    S. Shelah,A combinatorial problem; stability and order for models and theories in infinitary languages, Pac. J. Math.41 (1) (1972), 247–261.MATHGoogle Scholar
  21. 21.
    P. Frankl,Families of finite sets satisfying a union condition, Discrete Math.26 (1979), 111–118.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1983

Authors and Affiliations

  • N. Alon
    • 1
    • 2
  • V. D. Milman
    • 1
    • 2
  1. 1.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.Department of MathematicsTel Aviv UniversityTel AvivIsrael

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