Optimal realizations of generic 5-point metrics

Given a metric d on a finite set X, a realization of d is a triple (G,φ,w) consisting of a graph G = (V,E), a labeling φ : X → V , and a weighting w : E → R>0 such that for all x, y ∈ X the length of any shortest path in G between φ(x) and φ(y) equals d(x, y). Such a realization is called optimal if ‖G‖ := ∑ e∈E w(e) is minimal amongst all realizations of d. In this paper we will consider optimal realizations of generic 5-point metric spaces. In particular, we show that there is a canonical subdivision C of the metric fan of 5-point metrics into cones such that (i) every metric d in the interior of a cone C ∈ C has a unique optimal realization (G,φ,w), (ii) if d′ is also in the interior of C with optimal realization (G′, φ′, w′) then (G,φ) and (G′, φ′) are isomorphic as labeled graphs, and (iii) any labeled graph that underlies all optimal realizations of the metrics in the interior of some cone C ∈ C must belong to one of three isomorphism classes.