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 $\ell_\infty^n$.