Complementability of isometric copies of $\ell_1$ in transportation cost spaces

TL;DR

This paper proves that if a transportation cost space contains an isometric copy of , then it also contains a 1-complemented isometric copy, revealing a structural property of these spaces.

Contribution

It establishes the complementability of  in transportation cost spaces when such a copy exists, advancing understanding of their geometric structure.

Findings

If the transportation cost space contains , it contains a 1-complemented isometric copy.

The result applies to various contexts where transportation cost spaces are used.

Provides new insights into the structure of Lipschitz-free and Wasserstein 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 $\ell_1$.