# Generalized transportation cost spaces

## TL;DR

This paper explores the geometry of transportation cost spaces, proving new existence results and properties of metric spaces related to embeddings and norms, advancing understanding of their structure and limitations.

## Contribution

It introduces new existence results for specific metric spaces with unique embedding and norm properties, answering open questions in the field.

## Key findings

- Existence of infinite discrete metric spaces without isometric $\, ext{l}_1$ embeddings
- Characterization of locally finite metric spaces embedding only into Banach spaces containing $\, ext{l}_1$
- Identification of metric spaces where the double-point norm is not a norm

## Abstract

The paper is devoted to the geometry of transportation cost spaces and their generalizations introduced by Melleray, Petrov, and Vershik (2008). Transportation cost spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein $1$ spaces. In this work, the existence of metric spaces with the following properties is proved: (1) uniformly discrete infinite metric spaces transportation cost spaces on which do not contain isometric copies of $\ell_1$, this result answers a question raised by Cuth and Johanis (2017); (2) locally finite metric spaces which admit isometric embeddings only into Banach spaces containing isometric copies of $\ell_1$; (3) metric spaces for which the double-point norm is not a norm. In addition, it is proved that the double-point norm spaces corresponding to trees are close to $\ell_\infty^d$ of the corresponding dimension, and that for all finite metric spaces $M$, except a very special class, the infimum of all seminorms for which the embedding of $M$ into the corresponding seminormed space is isometric, is not a seminorm.