TY - JOUR

T1 - Linearly rigid metric spaces and the embedding problem

AU - Melleray, J.

AU - Petrov, F. V.

AU - Vershik, A. M.

PY - 2008/7/2

Y1 - 2008/7/2

N2 - 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.

AB - 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.

KW - Isometric embedding

KW - Kantorovich metric

KW - Linear rigidity

KW - Urysohn space

UR - http://www.scopus.com/inward/record.url?scp=45849148106&partnerID=8YFLogxK

U2 - 10.4064/fm199-2-6

DO - 10.4064/fm199-2-6

M3 - Article

AN - SCOPUS:45849148106

VL - 199

SP - 177

EP - 194

JO - Fundamenta Mathematicae

JF - Fundamenta Mathematicae

SN - 0016-2736

IS - 2

ER -