We consider the problem of isometric embedding of metric spaces into Banach spaces, and introduce and study the remarkable class of so-called linearly rigid metric spaces: these are the spaces that admit a unique, up to isometry, linearly dense isometric embedding into a Banach space. The first nontrivial example of such a space was given by R. Holmes; he proved that the universal Urysohn space has this property. We give a criterion of linear rigidity of a metric space, which allows us to give a simple proof of the linear rigidity of the Urysohn space and some other metric spaces. Various properties of linearly rigid spaces and related spaces are considered.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm199-2-6,
author = {J. Melleray and F. V. Petrov and A. M. Vershik},
title = {Linearly rigid metric spaces and the embedding problem},
journal = {Fundamenta Mathematicae},
volume = {201},
year = {2008},
pages = {177-194},
zbl = {1178.46016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm199-2-6}
}
J. Melleray; F. V. Petrov; A. M. Vershik. Linearly rigid metric spaces and the embedding problem. Fundamenta Mathematicae, Tome 201 (2008) pp. 177-194. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm199-2-6/