Type: Article
Publication Date: 2018-10-27
Citations: 8
DOI: https://doi.org/10.2140/ant.2018.12.1773
A pseudolength function defined on an arbitrary group G = (G, • , e, ( ) -1 ) is a map : G → [0, +∞) obeying (e) = 0, the symmetry property (x -1 ) = (x), and the triangle inequality (x y) ≤ (x) + (y) for all x, y ∈ G.We consider pseudolength functions which saturate the triangle inequality whenever x = y, or equivalently those that are homogeneous in the sense that (x n ) = n (x) for all n ∈ .ގWe show that this implies that ([x, y]) = 0 for all x, y ∈ G.This leads to a classification of such pseudolength functions as pullbacks from embeddings into a Banach space.We also obtain a quantitative version of our main result which allows for defects in the triangle inequality or the homogeneity property.