Transportation cost spaces and their embeddings into $L_1$, a Survey

TL;DR

This survey explores the properties of transportation cost spaces, also known as Lipschitzfree and Wasserstein spaces, focusing on their embeddings into L1 spaces, with added proofs from computational graph theory for completeness.

Contribution

It provides a comprehensive, self-contained overview of embeddings of transportation cost spaces into L1, including proofs from computational graph theory.

Findings

Characterization of bi-Lipschitz embeddings into L1

Conditions for linear embeddings into L1

Connections between transportation cost spaces and graph theory

Abstract

These notes present a basic survey on Transportation cost spaces (aka Lipschitzfree spaces, Wasserstein spaces) and their bi-Lipschitz and linear embeddings into $L_1$ spaces. To make these notes as self-contained as possible, we added the proofs of several relevant results from computational graph theory in the appendix.