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

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.