Geodesic
Uniqueness of Steiner minimal trees on boundaries
Sbornik. Mathematics, Tome 197 (2006) no. 9, pp. 1309-1340 Cet article a éte moissonné depuis la source Math-Net.Ru

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The following result is proved: there exists an open dense subset $U$ of $\mathbb R^{2n}$ such that each $P\in U$ (regarded as an enumerated subset of the standard Euclidean plane $\mathbb R^2$) is spanned by a unique Steiner minimal tree, that is, a shortest non-degenerate network. Several interesting consequences are also obtained: in particular, it is proved that each planar Steiner tree is planar equivalent to a Steiner minimal tree. Bibliography: 11 titles. 
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A. O. Ivanov; A. A. Tuzhilin. Uniqueness of Steiner minimal trees on boundaries. Sbornik. Mathematics, Tome 197 (2006) no. 9, pp. 1309-1340. http://geodesic.mathdoc.fr/item/SM_2006_197_9_a3/

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