Optimal realizations of generic five-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 five-point metric spaces. In particular, we show that there is a canonical subdivision C of the metric fan of five-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.