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

Question

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.

Answer 1

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