Israel Journal of Mathematics

, Volume 45, Issue 4, pp 265–280 | Cite as

Embedding ofl k in finite dimensional Banach spaces

  • N. Alon
  • V. D. Milman


Letx 1,x 2, ...,x n ben unit vectors in a normed spaceX and defineM n =Ave{‖Σ i=1 n ε1 x i ‖: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 {x i } i∈A is 16M n -isomorphic to the natural basis ofl k . This result implies a significant improvement of the known results concerning embedding ofl k in finite dimensional Banach spaces. We also prove that for every ∈>0 there exists a constantC(∈) such that every normed spaceX n of dimensionn either contains a (1+∈)-isomorphic copy ofl 2 m for somem satisfying ln lnm≧1/2 ln lnn or contains a (1+∈)-isomorphic copy ofl k for somek satisfying ln lnk>1/2 ln lnnC(∈). These results follow from some combinatorial properties of vectors with ±1 entries.


Normed Space Triangle Inequality Isomorphic Copy Combinatorial Result Pairwise Disjoint Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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.zbMATHCrossRefMathSciNetGoogle 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.zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    B. Bollobás,Extremal Graph Theory, Academic Press, London and New York, 1978, pp. 52–53.zbMATHGoogle 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.zbMATHGoogle 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.zbMATHCrossRefGoogle Scholar
  8. 8.
    T. Figiel and N. Tomczak-Jaegermann,Projections onto Hilbertian subspaces of Banach spaces, Isr. J. Math.33 (1979), 155–171.zbMATHCrossRefMathSciNetGoogle 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.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    D. P. Giesy,On a convexity condition in normed linear spaces, Trans. Am. Math. Soc.125 (1966), 114–146.zbMATHCrossRefMathSciNetGoogle 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.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    M. G. Karpovsky and V. Milman,Coordinate density of sets of vectors, Discrete Math.24 (1978), 177–184.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    D. R. Lewis,Finite dimensional subspaces of L p, Studia Math.63 (1978), 207–212.zbMATHMathSciNetGoogle 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.zbMATHMathSciNetGoogle 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.zbMATHMathSciNetGoogle Scholar
  17. 17.
    V. Milman,Some remarks about embedding of l 1k in finite dimensional spaces, Isr. J. Math.43 (1982), 129–138.zbMATHCrossRefMathSciNetGoogle 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.zbMATHCrossRefMathSciNetGoogle 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.zbMATHGoogle Scholar
  21. 21.
    P. Frankl,Families of finite sets satisfying a union condition, Discrete Math.26 (1979), 111–118.zbMATHCrossRefMathSciNetGoogle 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

Personalised recommendations