Abstract
Metric embeddings are central to metric theory and its applications. Here we consider embeddings of a different sort: maps from a set to subsets of a metric space so that distances between points are approximated by minimal distances between subsets. Our main result is a characterization of when a set of distances 𝒹(𝓍,𝓎) between elements in a set 𝒳 have a subtree representation, a real tree 𝘛 and a collection {S𝓍}𝓍∈𝒳 of subtrees of ~𝘛 such that 𝒹(𝓍,𝓎) equals the length of the shortest path in ~𝘛 from a point in S𝓍 to a point in S𝓎 for all 𝓍,𝓎 ∈ 𝒳. The characterization was first established for finite 𝒳 by Hirai (2006) using a tight span construction defined for distance spaces, metric spaces without the triangle inequality. To extend Hirai's result beyond finite 𝒳 we establish fundamental results of tight span theory for general distance spaces, including the surprising observation that the tight span of a distance space is hyperconvex. We apply the results to obtain the first characterization of when a diversity - a generalization of a metric space which assigns values to all finite subsets of 𝒳, not just to pairs - has a tight span which is tree-like.