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I need to use the following well-known result in my paper:

Let $X$ be a set of $n$ points in $\mathbb{R}^d$. Then $(X,\ell_2^d)$ embeds isometrically in $\ell_p^\binom{n}{2}$ for all $p \geq 1$.

What is the best reference to cite for this? I found a result which is nearly identical (but doesn't include the $\binom{n}{2}$ dimension bound) in some lecture notes from Michel Goemans, but I'm (a) unsure if I can cite lecture notes in a scientific publication, and (b) still in need of an $O(n^k)$ bound on the dimension.

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    $\begingroup$ Probably still not optimal, but this appears as Exercise 15.5.5 in Matousek’s “Lectures on Discrete Geometry.” $\endgroup$ – J.G May 14 at 6:27
  • $\begingroup$ @Elliot Gorokhovsky - Perhaps you will find some of the discussion here useful: iuuk.mff.cuni.cz/~koucky/LBCAD/papers/CubeAutomorphism.pdf $\endgroup$ – Avi Tal May 14 at 6:44
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This paper by Keith Ball seems to be what you are looking for:

Ball, Keith. "Isometric embedding in $\ell_p$-spaces." European Journal of Combinatorics 11.4 (1990): 305-311.

Link to the paper here: https://www.sciencedirect.com/science/article/pii/S019566981380131X

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  • $\begingroup$ Thanks! This works when combined with the Goemans lecture notes. $\endgroup$ – Elliot Gorokhovsky Jun 14 at 19:29

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