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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.

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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.

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