The Banach Mazur distance between n-dimensional space and $ \ell_{\infty} ^n$

Question

Let $X$ be an $n$-dimensinal space. Is the Banach-Mazur distance $d(X,\ell_{\infty}^n)$ less than or equal to $n$? Is $d(X,\ell_{\infty}^n)$ less than or equal to some constant $C(n)$ depending only on $n$?

Answer 1

It is well known in narrow circles that $$ d(Z,\ell_2^n)\leq\sqrt{n}\tag{1} $$ Furthermore it is easy to see straight from the definition that $$ d(U, V)\leq d(U,W)d(W,V)\tag{2} $$ Now set $U=X$, $V=\ell_\infty^n$, $W=\ell_2^n$ and apply $(1)$.