# Isometric copies of $\ell_\infty^n$ and $\ell_1^n$ in transportation cost spaces on finite metric spaces

## TL;DR

This paper investigates the geometric structure of transportation cost spaces on finite metric spaces, demonstrating the presence of isometric copies of classical Banach spaces like ^n and _, revealing their rich geometric properties.

## Contribution

It establishes the existence of isometric ^n and _ copies within transportation cost spaces on finite metric spaces, providing new insights into their geometric structure.

## Key findings

- Transportation cost space on a 2n-element metric space contains a 1-complemented isometric ^n.
- Existence of a finite metric space whose transportation cost space contains an isometric _.
- Transportation cost spaces include classical Banach spaces as isometric subspaces.

## Abstract

Main results: (a) If a metric space contains $2n$ elements, the transportation cost space on it contains a $1$-complemented isometric copy of $\ell_1^n$. (b) An example of a finite metric space whose transportation cost space contains an isometric copy of $\ell_\infty^4$. Transportation cost spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein $1$ spaces.