Israel Journal of Mathematics

, Volume 45, Issue 4, pp 265–280

Embedding ofl k in finite dimensional Banach spaces

Authors

  • N. Alon
    • Institute of MathematicsThe Hebrew University of Jerusalem
    • Department of MathematicsTel Aviv University
  • V. D. Milman
    • Institute of MathematicsThe Hebrew University of Jerusalem
    • Department of MathematicsTel Aviv University
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

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.

Copyright information

© Hebrew University 1983