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Given $n$, is there a $\big(1+C(n)\big)$-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 $p>2$.

We have an easy isometric embedding in the reverse direction though, $l^n_1 \to l^{n^2}_{\infty}$.

Given $n$, is there a $C(n)$-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 $p>2$.

We have an easy isometric embedding in the reverse direction though, $l^n_1 \to l^{2^n}_{\infty}$.

Given $n$, is there a $\big(1+C(n)\big)$-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 $p>2$.

Given $n$, is there a $\big(1+C(n)\big)$-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 $p>2$.

We have an easy isometric embedding in the reverse direction though, $l^n_1 \to l^{n^2}_{\infty}$.

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 $p>2$.

Given $n$, is there a $\big(1+C(n)\big)$-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 $p>2$.