Embedding of l(Formula presented.) in finite dimensional Banach spaces

N. Alon, V. D. Milman

Research output: Contribution to journalArticle

20 Scopus citations

Abstract

Let x1, x2, ..., xn be n unit vectors in a normed space X and define Mn=Ave{‖Σ(Formula presented.)ε1xi‖:ε1=±1}. We prove that there exists a set A⊂{1, ..., n} of cardinality(Formula presented.) such that {xi}i∈A is 16 Mn-isomorphic to the natural basis of l(Formula presented.). This result implies a significant improvement of the known results concerning embedding of l(Formula presented.) in finite dimensional Banach spaces. We also prove that for every ∈>0 there exists a constant C(∈) such that every normed space Xn of dimension n either contains a (1+∈)-isomorphic copy of l(Formula presented.) for some m satisfying ln ln m≧1/2 ln ln n or contains a (1+∈)-isomorphic copy of l(Formula presented.) for some k satisfying ln ln k>1/2 ln ln n−C(∈). These results follow from some combinatorial properties of vectors with ±1 entries.

Original languageEnglish (US)
Pages (from-to)265-280
Number of pages16
JournalIsrael Journal of Mathematics
Volume45
Issue number4
DOIs
StatePublished - Dec 1983
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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