### Abstract

It is shown that for every K > 0 and ε ∈ (0, 1/2) there exist N = N(K) ∈ N and D = D(K, ε) ∈ (1,∞) with the following properties. For every metric space (X, d) with doubling constant at most K, themetric space (X, d^{1-ε}) admits a bi-Lipschitz embedding into R^{N} with distortion at most D. The classical Assouad embedding theorem makes the same assertion, but with N →∞ as ε → 0.

Original language | English (US) |
---|---|

Pages (from-to) | 1123-1142 |

Number of pages | 20 |

Journal | Revista Matematica Iberoamericana |

Volume | 28 |

Issue number | 4 |

DOIs | |

State | Published - 2012 |

### Fingerprint

### Keywords

- Assouad's theorem
- Doubling metric spaces

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Revista Matematica Iberoamericana*,

*28*(4), 1123-1142. https://doi.org/10.4171/rmi/706

**Assouad's theorem with dimension independent of the snowflaking.** / Naor, Assaf; Neiman, Ofer.

Research output: Contribution to journal › Article

*Revista Matematica Iberoamericana*, vol. 28, no. 4, pp. 1123-1142. https://doi.org/10.4171/rmi/706

}

TY - JOUR

T1 - Assouad's theorem with dimension independent of the snowflaking

AU - Naor, Assaf

AU - Neiman, Ofer

PY - 2012

Y1 - 2012

N2 - It is shown that for every K > 0 and ε ∈ (0, 1/2) there exist N = N(K) ∈ N and D = D(K, ε) ∈ (1,∞) with the following properties. For every metric space (X, d) with doubling constant at most K, themetric space (X, d1-ε) admits a bi-Lipschitz embedding into RN with distortion at most D. The classical Assouad embedding theorem makes the same assertion, but with N →∞ as ε → 0.

AB - It is shown that for every K > 0 and ε ∈ (0, 1/2) there exist N = N(K) ∈ N and D = D(K, ε) ∈ (1,∞) with the following properties. For every metric space (X, d) with doubling constant at most K, themetric space (X, d1-ε) admits a bi-Lipschitz embedding into RN with distortion at most D. The classical Assouad embedding theorem makes the same assertion, but with N →∞ as ε → 0.

KW - Assouad's theorem

KW - Doubling metric spaces

UR - http://www.scopus.com/inward/record.url?scp=84876802613&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84876802613&partnerID=8YFLogxK

U2 - 10.4171/rmi/706

DO - 10.4171/rmi/706

M3 - Article

VL - 28

SP - 1123

EP - 1142

JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

SN - 0213-2230

IS - 4

ER -