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Linearly rigid metric spaces and the embedding problem. (English) Zbl 1178.46016

This very interesting paper deals with the so-called linearly rigid spaces. A metric space \(M\) is linearly rigid if there is a unique isometric embedding of \(M\) into a linearly dense subset of a Banach space \(X\), up to isometry, meaning that the linear span of the set is dense. It is not obvious that nontrivial linearly rigid spaces exist, but M.R.Holmes showed in [Fundam.Math.140, No.3, 199–223 (1992; Zbl 0772.54022)] that the Urysohn space is linearly rigid, where we recall that the Urysohn space is the unique Polish space which is isometrically universal for Polish spaces and ultra-homogeneous.
The main result of this paper is a characterization of linearly rigid spaces which gives an easy proof of Holmes’s result, but also provides more examples such as the rational and integral universal spaces. The characterization can be expressed in terms of compatible norms on the space \(V_{0}(M)\) of measures on \(M\) with finite support and total mass zero, where a norm is called compatible if the norm of the difference of two Dirac measures is the distance between the points. Linearly rigid spaces are shown to be those for which the Kantorovich-Rubinstein norm is the unique compatible norm.
A number of interesting problems are mentioned. For instance, it would be nice to identify the Banach space generated by the Urysohn space: a natural candidate is Gurarii’s space. This paper serves therefore also as an attractive work program.

MSC:

46B85 Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science
54E35 Metric spaces, metrizability
46B04 Isometric theory of Banach spaces
51F10 Absolute spaces in metric geometry

Citations:

Zbl 0772.54022
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