Linearly rigid metric spaces and the embedding problem

J. Melleray ¹   ; F. V. Petrov ²   ; A. M. Vershik ²

¹ UFR de Mathématiques Université Claude Bernard Lyon 1 43 boulevard du 11 novembre 1918 69622 Villeurbanne Cedex, France
² Steklov Mathematical Institute at St. Petersburg Fontanka 27 St. Petersburg 191023, Russia

Fundamenta Mathematicae, Tome 199 (2008) no. 2, pp. 177-194

Résumé

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.

Détail

DOI : 10.4064/fm199-2-6

Keywords: consider problem isometric embedding metric spaces banach spaces introduce study remarkable class so called linearly rigid metric spaces these spaces admit unique isometry linearly dense isometric embedding banach space first nontrivial example space given holmes proved universal urysohn space has property criterion linear rigidity metric space which allows simple proof linear rigidity urysohn space other metric spaces various properties linearly rigid spaces related spaces considered