<p>This paper by Keith Ball seems to be what you are looking for:</p>

<blockquote>
  <p>Ball, Keith. "Isometric embedding in <span class="math-container">$\ell_p$</span>-spaces." 
  European Journal of Combinatorics 11.4 (1990): 305-311.</p>
</blockquote>

<p>Link to the paper here:
<a href="https://www.sciencedirect.com/science/article/pii/S019566981380131X" rel="nofollow noreferrer">https://www.sciencedirect.com/science/article/pii/S019566981380131X</a></p>

CC BY-SA 4.0
2020-06-14T19:29:03.597Z
Thanks! This works when combined with the Goemans lecture notes.
Elliot Gorokhovsky
CC BY-SA 4.0
2020-06-12T23:10:04.737Z
2020-06-12T23:10:04.737Z
Cyrus Rashtchian
<p>I need to use the following well-known result in my paper:</p>

<blockquote>
  <p>Let <span class="math-container">$X$</span> be a set of <span class="math-container">$n$</span> points in <span class="math-container">$\mathbb{R}^d$</span>. Then <span class="math-container">$(X,\ell_2^d)$</span> embeds isometrically in <span class="math-container">$\ell_p^\binom{n}{2}$</span> for all <span class="math-container">$p \geq 1$</span>.</p>
</blockquote>

<p>What is the best reference to cite for this? I found a result which is nearly identical (but doesn't include the <span class="math-container">$\binom{n}{2}$</span> dimension bound) in some <a href="http://www-math.mit.edu/~goemans/18409-2006/lec1.pdf" rel="nofollow noreferrer">lecture notes</a> 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 <span class="math-container">$O(n^k)$</span> bound on the dimension.</p>

CC BY-SA 4.0
2020-05-14T06:27:55.830Z
Probably still not optimal, but this appears as Exercise 15.5.5 in Matousek’s “Lectures on Discrete Geometry.”
Jason Gaitonde
CC BY-SA 4.0
2020-05-14T06:44:21.053Z
@Elliot Gorokhovsky - Perhaps you will find some of the discussion here useful:    https://iuuk.mff.cuni.cz/~koucky/LBCAD/papers/CubeAutomorphism.pdf
Avi Tal
<reference-request><cg.comp-geom><metric-spaces><embeddings>
Citation for isometric embeddability of $\ell_2$ into $\ell_p^\binom{n}{2}$ for $p \geq 1$?
Elliot Gorokhovsky
# Citation for isometric embeddability of $\ell_2$ into $\ell_p^\binom{n}{2}$ for $p \geq 1$?

(Asked by: username_0 on 2020-05-13)

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](http://www-math.mit.edu/~goemans/18409-2006/lec1.pdf) 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.

**Tags:** reference-request, cg.comp-geom, metric-spaces, embeddings
> **Comments:**
> **username_1**: Probably still not optimal, but this appears as Exercise 15.5.5 in Matousek’s “Lectures on Discrete Geometry.”  
> **username_2**: @Elliot Gorokhovsky - Perhaps you will find some of the discussion here useful:    https://iuuk.mff.cuni.cz/~koucky/LBCAD/papers/CubeAutomorphism.pdf  

---

## 1 answers

### Answer by username_3 on 2020-06-12. Score: 4 | Accepted answer

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>

> **Comments:**
> **username_0**: Thanks! This works when combined with the Goemans lecture notes.  

---

