Linearly rigid metric spaces and the embedding problem

• Autores: Julien Melleray, F. V. Petrov, A. M. Vershik
• Localización: Fundamenta mathematicae, ISSN 0016-2736, Vol. 199, Nº 2, 2008, págs. 177-194
• Idioma: inglés
• DOI: 10.4064/fm199-2-6
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• Resumen

  • 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.