# Citation for isometric embeddability of $\ell_2$ into $\ell_p^\binom{n}{2}$ for $p \geq 1$?

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.

• Probably still not optimal, but this appears as Exercise 15.5.5 in Matousek’s “Lectures on Discrete Geometry.” May 14, 2020 at 6:27
• @Elliot Gorokhovsky - Perhaps you will find some of the discussion here useful: iuuk.mff.cuni.cz/~koucky/LBCAD/papers/CubeAutomorphism.pdf May 14, 2020 at 6:44

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