Banach–Mazur Distance from $\ell_p^3$ to $\ell_\infty^3$

The maximum of the Banach–Mazur distance $d_{BM}^M(X,\ell_\infty^n)$, where $X$ ranges over the set of all $n$-dimensional real Banach spaces, is difficult to compute. In fact, it is even not easy to find the maximum of $d_{BM}^M(\ell_p^n,\ell_\infty^n)$ over all $p\in [1,\infty]$. We prove that $d_{BM}^M(\ell_p^3,\ell_\infty^3)\leq 9/5$ for all $p\in[1,\infty]$. As an application, the following result related to Borsuk's partition problem in Banach spaces is obtained: any subset $A$ of $\ell_p^3$ having diameter $1$ is a union of $8$ subsets of $A$ whose diameters are at most $0.9$.