Uniqueness of Steiner minimal trees on boundaries in general position

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.
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