### Abstract

The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of largest cardinality which can be embedded with a given distortion in Hilbert space. We provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion. Our main theorem states that for any ε > 0, every n point metric space contains a subset of size at least n ^{1-ε} which is embeddable in Hilbert space with O ( log(1/ε)/ε) distortion. The bound on the distortion is tight up to the log(1/ε) factor. We further include a comprehensive study of various other aspects of this problem.

Original language | English (US) |
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Pages (from-to) | 643-709 |

Number of pages | 67 |

Journal | Annals of Mathematics |

Volume | 162 |

Issue number | 2 |

DOIs | |

State | Published - Sep 1 2005 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Annals of Mathematics*,

*162*(2), 643-709. https://doi.org/10.4007/annals.2005.162.643