Three-way Metrics: Axiomatization and Properties

Abstract This note deals with three-way dissimilarities only defined on unordered triples. They are in relationship with two-way dissimilarities via an Lp-transformation and a particular attention is paid to the perimeter model (p = 1). In that case, a six-point condition is established. Following the basic papers of Joly-Le Calve (J. of Classification, 1995) and Heiser-Bennani (J. of Math. Psychology, 1997), axiomatizations for metricity, ultrametricity and tree-metricity are discussed. Every three-way dissimilarity of perimeter type deriving from a two-way metric is a strong metric, sensus Heiser-Bennani. However, in our context the converse is no longer true and a second six-point characterization is established. Ultrametricity and tree-metricity are developed in the same spirit. For p equal to the infinity, every transform of a two-way ultrametric is a so-called three-way strong ultrametric. Moreover, a property based on the subdominant ultrametric specifies those three-way ultrametrics. Embeddability in a tree structure seems to require the perimeter model. So, we define a three-way tree metric as a metric of perimeter type deriving from a two-way tree metric. Still, a three-way dissimilarity fulfills this definition if and only if its restriction to every subset of six points does, and a counter-example shows that such a condition is sharp. Equivalently, it is of perimeter type and obeys a five-point property, similar to the four-point one established for two-way tree metricity. Finally, every ultrametric of perimeter type is in this class.