Characterization of metric spaces whose free space is isometric to $\ell_1$

TL;DR

This paper characterizes metric spaces whose Lipschitz free space is isometric to , showing ultrametric spaces do not have this property and providing bounds on Banach-Mazur distance in finite cases.

Contribution

It provides a complete characterization of metric spaces with Lipschitz free space isometric to , including ultrametric spaces and finite case bounds.

Findings

Ultrametric spaces' free spaces are not isometric to 

Lower bounds for Banach-Mazur distance in finite cases

Characterization of spaces with free space isometric to 

Abstract

We characterize metric spaces whose Lipschitz free space is isometric to $\ell_1$. In particular, the Lipschitz free space over an ultrametric space is not isometric to $\ell_1(\Gamma)$ for any set $\Gamma$. We give a lower bound for the Banach-Mazur distance in the finite case.