Israel Journal of Mathematics

, Volume 45, Issue 4, pp 265–280

Embedding oflk in finite dimensional Banach spaces

  • N. Alon
  • V. D. Milman
Article

DOI: 10.1007/BF02804012

Cite this article as:
Alon, N. & Milman, V.D. Israel J. Math. (1983) 45: 265. doi:10.1007/BF02804012

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.

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