### Abstract

Let x_{1}, x_{2}, ..., x_{n} be n unit vectors in a normed space X and define M_{n}=Ave{‖Σ(Formula presented.)ε_{1}x_{i}‖:ε_{1}=±1}. We prove that there exists a set A⊂{1, ..., n} of cardinality(Formula presented.) such that {x_{i}}_{i∈A} is 16 M_{n}-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 X_{n} 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 language | English (US) |
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Pages (from-to) | 265-280 |

Number of pages | 16 |

Journal | Israel Journal of Mathematics |

Volume | 45 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1983 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Israel Journal of Mathematics*,

*45*(4), 265-280. https://doi.org/10.1007/BF02804012