### Abstract

This paper is devoted to the study of quotients of finite metric spaces. The basic type of question we ask is: Given a finite metric space M and α≥1, what is the largest quotient of (a subset of) M which well embeds into Hilbert space. We obtain asymptotically tight bounds for these questions, and prove that they exhibit phase transitions. We also study the analogous problem for embeddings into ℓ_{p}, and the particular case of the hypercube.

Original language | English (US) |
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Pages (from-to) | 451-494 |

Number of pages | 44 |

Journal | Advances in Mathematics |

Volume | 189 |

Issue number | 2 |

DOIs | |

State | Published - Dec 20 2004 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*189*(2), 451-494. https://doi.org/10.1016/j.aim.2003.12.001

**Euclidean quotients of finite metric spaces.** / Mendel, Manor; Naor, Assaf.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 189, no. 2, pp. 451-494. https://doi.org/10.1016/j.aim.2003.12.001

}

TY - JOUR

T1 - Euclidean quotients of finite metric spaces

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2004/12/20

Y1 - 2004/12/20

N2 - This paper is devoted to the study of quotients of finite metric spaces. The basic type of question we ask is: Given a finite metric space M and α≥1, what is the largest quotient of (a subset of) M which well embeds into Hilbert space. We obtain asymptotically tight bounds for these questions, and prove that they exhibit phase transitions. We also study the analogous problem for embeddings into ℓp, and the particular case of the hypercube.

AB - This paper is devoted to the study of quotients of finite metric spaces. The basic type of question we ask is: Given a finite metric space M and α≥1, what is the largest quotient of (a subset of) M which well embeds into Hilbert space. We obtain asymptotically tight bounds for these questions, and prove that they exhibit phase transitions. We also study the analogous problem for embeddings into ℓp, and the particular case of the hypercube.

UR - http://www.scopus.com/inward/record.url?scp=4344706090&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4344706090&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2003.12.001

DO - 10.1016/j.aim.2003.12.001

M3 - Article

VL - 189

SP - 451

EP - 494

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 2

ER -