Linearly rigid metric spaces and the embedding problem

Tom 199 / 2008

J. Melleray, F. V. Petrov, A. M. Vershik Fundamenta Mathematicae 199 (2008), 177-194 MSC: 46B04, 51F10. DOI: 10.4064/fm199-2-6

Streszczenie

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.

Autorzy

  • J. MellerayUFR de Mathématiques
    Université Claude Bernard Lyon 1
    43 boulevard du 11 novembre 1918
    69622 Villeurbanne Cedex, France
    e-mail
  • F. V. PetrovSteklov Mathematical Institute at St. Petersburg
    Fontanka 27
    St. Petersburg 191023, Russia
    e-mail
  • A. M. VershikSteklov Mathematical Institute at St. Petersburg
    Fontanka 27
    St. Petersburg 191023, Russia
    e-mail

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