Antipodal Metrics and Split Systems

Recall that a metric d on a finite set X is called antipodal if there exists a map σ: X → X: x ∣ → __x so that d(x, __ x) = d(x,y ) + d(y, __ x) holds for all x,y ∈ X. Antipodal metrics canonically arise as metrics induced on specific weighted graphs, although their abundance becomes clearer in light of the fact that any finite metric space can be isometrically embedded in a more or less canonical way into an antipodal metric space called its full antipodal extension. s paper, we examine in some detail antipodal metrics that are, in addition, totally split decomposable. In particular, we give an explicit characterization of such metrics, and prove that—somewhat surprisingly—the full antipodal extension of a proper metricd on a finite set X is totally split decomposable if and only if d is linear or#X = 3 holds.