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 ; 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm187-3-3,
author = {Jeremy T. Tyson},
title = {Bi-Lipschitz embeddings of hyperspaces of compact sets},
journal = {Fundamenta Mathematicae},
volume = {185},
year = {2005},
pages = {229-254},
zbl = {1096.54007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm187-3-3}
}
Jeremy T. Tyson. Bi-Lipschitz embeddings of hyperspaces of compact sets. Fundamenta Mathematicae, Tome 185 (2005) pp. 229-254. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm187-3-3/