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 7→ 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. In this paper, we examine is 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 metric d on a finite set X is totally split decomposable if and only if either #X = 3 or d is linear.