Bi-Lipschitz embeddings of hyperspaces of compact sets

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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 languageEnglish (US)
Pages (from-to)229-254
Number of pages26
JournalFundamenta Mathematicae
Volume187
Issue number3
DOIs
StatePublished - 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

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