Bi-Lipschitz embeddings of hyperspaces of compact sets

Jeremy T. Tyson¹

¹ Department of Mathematics University of Illinois 1409 West Green Street Urbana, IL 61801, U.S.A.

Fundamenta Mathematicae, Tome 187 (2005) no. 3, pp. 229-254.

Résumé

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 $\ell^2$. If $X$ is a countable compact metric space containing at most $n$ nonisolated points, there is a Lipschitz embedding of $K(X)$ in $\mathbb R^{n+1}$; 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.

Détail

DOI : 10.4064/fm187-3-3

Keywords: study bi lipschitz embedding problem metric compacta hyperspaces observe compacta hyperspace separable uniformly disconnected metric space admits bi lipschitz embedding ell countable compact metric space containing nonisolated points there lipschitz embedding mathbb presence additional convergence condition embedding may chosen bi lipschitz contrast hyperspace unit interval contains bi lipschitz copy certain self similar doubling series parallel graph studied laakso lang plaut lee mendel naor consequently admits bi lipschitz embedding uniformly convex banach space schori west proved homeomorphic hilbert cube while hohti showed bi lipschitz equivalent variety metric hilbert cubes