Bi-Lipschitz embeddings of hyperspaces of compact sets

• Autorzy: Jeremy T. Tyson
• Języki publikacji: 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.

Kategorie tematyczne

• 46B20: Geometry and structure of normed linear spaces
• 05C10: Planar graphs; geometric and topological aspects of graph theory
• 26A18: Iteration
• 54B20: Hyperspaces
• 26A16: Lipschitz (H\"older) classes
• 28A80: Fractals

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2005
• Tom: 187
• Numer: 3
• Strony: 229-254

Daty

wydano

2005

Twórcy

autor

Jeremy T. Tyson

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

Identyfikatory

DOI

10.4064/fm187-3-3