Complementability of isometric copies of ${\ell }_1$ in transportation cost spaces

Abstract

This work aims to establish new results pertaining to the structure of transportation cost spaces. Due to the fact that those spaces were studied and applied in various contexts, they have also become known under different names such as Arens-Eells spaces, Lipschitz-free spaces, and Wasserstein spaces. The main outcome of this paper states that if a metric space $X$ is such that the transportation cost space on $X$ contains an isometric copy of ${\ell }_1$, then it contains a $1$-complemented isometric copy of ${\ell }_1$.