Banach spaces with shortest network length depending only on pairwise distances between points

L. Sh. Burusheva

Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

Abstract: For a real Banach space realising shortest networks for all finite subsets, we prove that a necessary and sufficient condition for the shortest network length to be expressed as a function only of pairwise distances between its points is that the space is either predual to $L_1$ or a Hilbert space. We give a characterization of spaces predual to $L_1$ and Hilbert spaces in terms of shortest networks.
Bibliography: 23 titles.

Keywords: Banach space, shortest network, Steiner point, Lindenstrauss spaces.

UDC: 517.982.256+515.124.4

MSC: 46B04, 46B20

Received: 22.09.2017 and 20.11.2018

DOI: 10.4213/sm9012

 English version: Sbornik: Mathematics, 2019, 210:3, 297–309

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