### Abstract

Let X be a normed space that satisfies the Johnson-Lindenstrauss lemma (J-L lemma, in short) in the sense that for any integer n and any x_{1}, . . . , x_{n} ∈ X there exists a linear mapping L:X → F, where F ⊆ X is a linear subspace of dimension O(log n), such that ∥x _{i} - x_{j}∥ ≤ ∥L(x_{i}) - L(x _{j})∥ ≤ O(1)·∥x_{i} - x_{j}∥ for all i, j ∈ {1, . . . , n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion 2^{2O(log * n)}. On the other hand, we show that there exists a normed space Y which satisfies the J-L lemma, but for every n there exists an n-dimensional subspace E_{n} ⊆ Y whose Euclidean distortion is at least 2^{Ω(α(n))}, where α is the inverse Ackermann function.

Original language | English (US) |
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Title of host publication | Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms |

Publisher | Association for Computing Machinery |

Pages | 885-891 |

Number of pages | 7 |

ISBN (Print) | 9780898716801 |

DOIs | |

State | Published - 2009 |

Externally published | Yes |

Event | 20th Annual ACM-SIAM Symposium on Discrete Algorithms - New York, NY, United States Duration: Jan 4 2009 → Jan 6 2009 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Other

Other | 20th Annual ACM-SIAM Symposium on Discrete Algorithms |
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Country | United States |

City | New York, NY |

Period | 1/4/09 → 1/6/09 |

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

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## Cite this

*Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms*(pp. 885-891). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms). Association for Computing Machinery. https://doi.org/10.1137/1.9781611973068.96