On the Banach–Mazur distance from $\ell_{1}^{n}$

We study the Banach–Mazur distance of an $ n $-dimensional Banach space from $\ell_{1}^n$, and obtain a useful formula for it. We explore several applications of this distance formula. In particular, we give an explicit characterization of the operators attaining the Banach–Mazur distance between $\ell_{1}^n $ and $ \ell_{2}^n$. We also present an analytic approach to study the geometric aspects of the Banach–Mazur distance, which leads to improvements of some earlier results in this context.