Optimal realizations of generic five-point metrics

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