On relations between transportation cost spaces and $\ell_1$

TL;DR

This paper investigates the structure of transportation cost spaces, providing conditions for when they contain an isometric copy of  and describing their structure for finite graphs, advancing understanding of these spaces in metric geometry.

Contribution

It establishes a necessary and sufficient condition for  in transportation cost spaces and generalizes the description of these spaces for finite graphs.

Findings

Characterized when transportation cost spaces contain 

Described transportation cost space of finite graphs as a quotient of 

Answered open questions on specific metric spaces

Abstract

The present paper deals with some structural properties of transportation cost spaces, also known as Arens-Eells spaces, Lipschitz-free spaces and Wasserstein spaces. The main results of this work are: (1) A necessary and sufficient condition on an infinite metric space $M$, under which the transportation cost space on $M$ contains an isometric copy of $\ell_1$. The obtained condition is applied to answer the open questions asked by C\'uth and Johanis (2017) concerning several specific metric spaces. (2) The description of the transportation cost space of a weighted finite graph $G$ as the quotient $\ell_1(E(G))/Z(G)$, where $E(G)$ is the edge set and $Z(G)$ is the cycle space of $G$. This is a generalization of the previously known result to the case of any finite metric space.