Geodesic
An open family of sets that have several minimal fillings
Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 2, pp. 153-156.

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Minimal fillings of $n$-point metric spaces were introduced by Ivanov and Tuzhilin. It was thought that “for almost all sets” in some sense such a filling is unique. Here we introduce a counterexample to this hypothesis. 
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Z. N. Ovsyannikov. An open family of sets that have several minimal fillings. Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 2, pp. 153-156. http://geodesic.mathdoc.fr/item/FPM_2013_18_2_a11/

[1] Erëmin A. Yu., “Formula vesa minimalnogo zapolneniya konechnogo metricheskogo prostranstva”, Mat. sb., 204:9 (2013), 51–72 | DOI | Zbl

[2] Ivanov A. O., Tuzhilin A. A., “Odnomernaya problema Gromova o minimalnom zapolnenii”, Mat. sb., 203:5 (2012), 65–118 | DOI | MR | Zbl