Geodesic
On the Existence of Shortest Networks in Banach Spaces
Matematičeskie zametki, Tome 94 (2013) no. 1, pp. 46-54.

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For dual spaces, and also for $L_1$, it is proved that every system of points in such a space admits a shortest network connecting the points. An example of a Banach space is presented in which, for every $n\ge 3$, there is a system of $n$ points which cannot be connected by a shortest network.
Keywords: Banach space, networks connecting given points, shortest network. 
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B. B. Bednov; N. P. Strelkova. On the Existence of Shortest Networks in Banach Spaces. Matematičeskie zametki, Tome 94 (2013) no. 1, pp. 46-54. http://geodesic.mathdoc.fr/item/MZM_2013_94_1_a3/

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