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• # Artykuł - szczegóły

## Fundamenta Mathematicae

2005 | 187 | 3 | 229-254

## Bi-Lipschitz embeddings of hyperspaces of compact sets

EN

### Abstrakty

EN
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 ℓ². 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.

229-254

wydano
2005

### Twórcy

autor
• Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, U.S.A.