The ℓ p-function on trees

A p-value of a sequence π = (x1, x2,⋯, xk) of elements of a finite metric space (X, d) is an element x for which ∑i=1kdp(x,xi) is minimum. The function ℓpwith domain the set of all finite sequences defined by ℓp(π) = {x: x is a p-value of π} is called the ℓp-function on X. The ℓp-functions with p = 1 and p = 2 are the well-studied median and mean functions respectively. In this article, the ℓp-function on finite trees is characterized axiomatically.