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.

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.

Reviewer: Clemente Zanco (Milano)