Isometric copies of $\ell_\infty^n$ in Lipschitz-free spaces over finite metric spaces

Abstract

For every $n\in\mathbb N$, we construct a finite metric subspace $M_n$ of $\ell_\infty^n$ such that the Lipschitz-free space $\mathcal F(M_n)$ contains a linear isometric copy of $\ell_\infty^n$. This answers a question posed by Khan, Mim, and Ostrovskii, who obtained examples in dimensions three and four. As a consequence, the finite-dimensional real Banach spaces that admit a linear isometric embedding into a Lipschitz-free space over a finite metric space are precisely the polyhedral spaces.