Linearly rigid metric spaces and the embedding problem
Volume 199 / 2008
Fundamenta Mathematicae 199 (2008), 177-194
MSC: 46B04, 51F10.
DOI: 10.4064/fm199-2-6
Abstract
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.
