The ℓ_p‐function on trees

Abstract

A p-value of a sequence π = (x 1 , x 2 , . . . , x k ) of elements of a finite metric space (X , d ) is an element x for whichThe function p with domain the set of all finite sequences defined by p (π) = {x : x is a p-value of π } is called the p -function on X . The p -functions with p = 1 and p = 2 are the well-studied median and mean functions respectively. In this article, the p -function on finite trees is characterized axiomatically.

Can We “Future-Proof” Consensus Trees?†

“…(2.3) holds for the main consensus methods mentioned so far is provided in the Appendix. Condition (2.3) corresponds to an "Invariance" property in [16].…”

The median function on Boolean lattices

“…Also in the discrete case the mean function was studied by Biagi [6] in her MA thesis where she found some relevant properties of this function on finite trees. These properties were the genesis for an axiomatic characterization of the mean function in [16] and for the p -function with p ≥ 2 in [17]. The characterization of the p -function gave a new characterization of the mean function.…”

Unimodal eccentricity in trees

“…We also prove unimodality of some functions that, to our knowledge, have not been known to have the Jordan property. A prototypical example is the ℓ p ‐function [7], for real p > 1.…”