Isometric copies of ${\ell }_{\infty }^n$ and ${\ell }_1^n$ in transportation cost spaces on finite metric spaces

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.