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An extension (V, d) of a metric space (S, μ) is a metric space with $${{S \subseteq V}}$$ and $${{d\mid{_S} = \mu}}$$ , and is said to be tight if there is no other extension (V, d′) of (S, μ) with d′ ≤ d . Isbell and Dress independently found that every tight extension embeds isometrically into a certain metrized polyhedral complex associated with (S, μ), called the tight span. This paper...
. A characterization is given to the distance between subtrees of a tree defined as the shortest path length between subtrees. This is a generalization of the four-point condition for tree metrics. For this, we use the theory of the tight span and obtain an extension of the famous result by Dress that a metric is a tree metric if and only if its tight span is a tree.
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