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Banach spaces that realize minimal fillings. (English. Russian original) Zbl 1327.46013
Sb. Math. 205, No. 4, 459-475 (2014); translation from Mat. Sb. 205, No. 3, 3-20 (2014).
Given a metric space $$(X,d)$$ and a finite subset $$M$$ of $$X$$, a network in $$X$$ spanning $$M$$ is a map $$\Gamma$$ from the set $$V$$ of the vertices of some connected graph to $$X$$ such that $$M \subset \Gamma(V)$$. The length of a network is defined in the natural way, starting from the edges of the graph under consideration. The main result of this paper is the following. A Banach space $$X$$ is a Lindenstrauss space (i.e., a predual of $$L_1$$) if and only if for each finite subset $$M$$ of $$X$$ there exists a shortest network in $$X$$ spanning $$M$$ and that network has the minimum possible length under all the isometric embeddings of $$M$$ into a metric space. This is equivalent to asking the same property to hold just for 4-points sets $$M$$.
The paper is pleasant to read. It contains interesting examples and is rich in references and connections with related subjects. For instance, it contains a direct proof of an isometric characterization of $$L_1$$ spaces via Steiner points (medians) of 3-point sets or via the metric segments determined by those sets. As the authors point out, that characterization can also be obtained in a nontrivial way via a result by A. Lima [Trans. Am. Math. Soc. 227, 1–62 (1977; Zbl 0347.46017)]. Finally, some interesting investigations are suggested.

##### MSC:
 46B04 Isometric theory of Banach spaces 05C12 Distance in graphs 54E35 Metric spaces, metrizability
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