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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
Article

Abstract

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.

Keywords

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.

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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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