Given $n$, is there a (1+C(n))-isomorphic embedding of $l^n_{\infty}$ into $l^m_1$ for sufficiently large m and $C(n)<<\log(n)$? 
For n=2 this can done with m=2.
There are some results about $(1+\epsilon)$-isometric embedding of $l^n_p$ into $l^m_1$ for $p\leq 2$ but I couldn't find anything for $p>2$.