Given an injective linear map $T$ between Banach spaces $X$ and $Y$, let 


\begin{equation} d(T) = \sup \left \{ \frac{||x||\_X}{||Tx||\_Y}: x \in X \mbox{ is nonzero } \right\} \cdot ||T||\_{\mathrm{op}} \end{equation}


Let \begin{equation} c(X,Y) = \inf \Bigl\{ d(T): T \mbox{ is an injective linear map from } X \mbox{ to }Y \Bigr \}. \end{equation}


Let $\ell^p\_n$ be $\mathbb{R}^n$ with the $p$-norm and let $\mathcal{H}$ be an infinite dimensional Hilbert space. I would like to know if $c(\ell^1\_n,\mathcal{H}) = O((\log n)^k)$ for some $k$. Alternatively, is it the case that $c(\ell^p\_n,\mathcal{H}) = O\_p((\log n)^{k(p)})$ for every $p > 1$?


There is a lot of literature about the Banach-Mazur distance but I have been unable to find information about linear embeddings (rather than isomorphisms). On the other hand, there is a well known result of Bourgain which asserts that a metric space with $n$ points can be embedded in $\mathcal{H}$ with distortion $O(\log n)$, but I don't know if this can be done in a uniform linear way.



Distortion of embedding in Hilbert space
You are asking what is the Banach-Mazur distance between $\ell\_p^n$ and the Hilbert space of dimension $n$. The answer is $n^{|1/p-1/2|}$. You can find this in many books; in particular, the book of Tomczak-Jaegermann. It does not change anything if you use the Lipschitz analogue of the Banach-Mazur distance. 


It is trickier to compute $c(\ell\_p^n,L\_r)$, which is the minimum of the Banach-Mazur distance from $\ell\_p^n$ to subspaces of $L\_r$. The order of magnitude is known for every $p$ and $r$ up to a constant that depends on $p$ and $r$, but for some $p$ and $r$ I think the exact values are not known.


