  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 already not easy to get the maximum of
$d_{BM}^M(\ell_p^n,\ell_\infty^n)$ for all $p\in [1,\infty]$. We prove that
$d_{BM}^M(\ell_p^3,\ell_\infty^3)\leq 9/5,~\forall p\in[1,\infty]$. 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
the union of $8$ subsets of $A$ whose diameters are at most $0.9$.
