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.

arXiv comment4 pages