Linearly rigid metric spaces and the embedding problem

Details Linearly rigid metric spaces and the embedding problem Linearly rigid metric spaces and the embedding problem

2006-11-02

arxiv.org/abs/math/0611049v4

Mathematics

Functional Analysis

23 pp. Ref.19

Scientific paper

We consider the problem of isometric embedding of metric spaces to the 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. The various properties of linearly rigid spaces and related spaces are considered.

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