### Abstract

We study the bi-Lipschitz embedding problem for metric compacta hyperspaces. We observe that the compacta hyperspace K(X) of any separable, uniformly disconnected metric space X admits a bi-Lipschitz embedding in ℓ ^{2}. If X is a countable compact metric space containing at most n nonisolated points, there is a Lipschitz embedding of K(X) in ℝ ^{n+1}; in the presence of an additional convergence condition, this embedding may be chosen to be bi-Lipschitz. By way of contrast, the hyperspace K([0,1]) of the unit interval contains a bi-Lipschitz copy of a certain self-similar doubling series-parallel graph studied by Laakso, Lang-Plaut, and Lee-Mendel-Naor, and consequently admits no bi-Lipschitz embedding into any uniformly convex Banach space. Schori and West proved that K([0,1]) is homeomorphic with the Hilbert cube, while Hohti showed that K([0,1]) is not bi-Lipschitz equivalent with a variety of metric Hilbert cubes.

Original language | English (US) |
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Pages (from-to) | 229-254 |

Number of pages | 26 |

Journal | Fundamenta Mathematicae |

Volume | 187 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1 2005 |

### Keywords

- Bi-lipschitz embedding
- Compacta hyperspace
- Iterated function system
- Round ball metric space
- Series-parallel graph

### ASJC Scopus subject areas

- Algebra and Number Theory