Bi-Lipschitz embeddings of hyperspaces of compact sets

Jeremy T. Tyson. "Bi-Lipschitz embeddings of hyperspaces of compact sets." Fundamenta Mathematicae 187.3 (2005): 229-254. <http://eudml.org/doc/283286>.

@article{JeremyT2005,
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 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.},
author = {Jeremy T. Tyson},
journal = {Fundamenta Mathematicae},
keywords = {bi-Lipschitz embedding; compacta hyperspace; iterated function system; round ball metric space; series-parallel graph},
language = {eng},
number = {3},
pages = {229-254},
title = {Bi-Lipschitz embeddings of hyperspaces of compact sets},
url = {http://eudml.org/doc/283286},
volume = {187},
year = {2005},
}

TY - JOUR
AU - Jeremy T. Tyson
TI - Bi-Lipschitz embeddings of hyperspaces of compact sets
JO - Fundamenta Mathematicae
PY - 2005
VL - 187
IS - 3
SP - 229
EP - 254
AB - 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.
LA - eng
KW - bi-Lipschitz embedding; compacta hyperspace; iterated function system; round ball metric space; series-parallel graph
UR - http://eudml.org/doc/283286
ER -