An axiomatic approach to location functions on finite metric spaces

A location function on a finite metric space (X,d) is a function on the set, X⁎, of all finite sequences of elements of X, to 2X\∅, which minimizes some criteria of remoteness. Axiomatic characterizations of these functions have, for the most part, been established only for very special cases. While McMorris, Mulder and Powers [F.R. McMorris, H.M. Mulder, R.C. Powers, “The median function on median graphs and semilattices,” Discrete Appl. Math., 101, (2000), 221–230] were able to characterize the median function on median graphs with three axioms, one of their axioms was very specific to the structure of median graphs. Recently, however, Mulder and Novick [H.M. Mulder, B.A. Novick, “A tight axiomatization of the median procedure on median graphs,” Discrete Appl. Math., 161, (2013), 838–846] characterized the median function for all median graphs using only three very natural axioms. These three axioms are meaningful in the more general context of finite metric spaces. In this work, we establish that these same three axioms are indeed independent and then we settle completely the question of interdependence among the collection of axioms involved in the above mentioned two characterizations, giving examples for all logically relevant cases. We introduce several new location functions and pose some questions.