Subtree Distances, Tight Spans and Diversities

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 $\{S_x\}_{x \in X}$ of subtrees of~$T$ such that $d(x,y)$ equals the length of the shortest path in~$T$ from a point in $S_x$ to a point in $S_y$ for all $x,y \in X$. The characterization was first established for {\em 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.