When is a subspace of ${\ell }_{\infty }^N$ isometrically isomorphic to ${\ell }_{\infty }^n$?

Abstract

It is shown in this note that one can decide whether an $n$-dimensional subspace of ${\ell }_{\infty }^N$ is isometrically isomorphic to ${\ell }_{\infty }^n$ by testing a finite number of determinental inequalities. As a byproduct, an elementary proof is provided for the fact that an $n$-dimensional subspace of ${\ell }_{\infty }^N$ with projection constant equal to one must be isometrically isomorphic to ${\ell }_{\infty }^n$.