I am looking for a reference to the bipartite version of the Schoenberg's criterion of embeddability into a Hilbert space. The Schoenberg criterion is formulated as Proposition 8.5(ii) of the book "[Geometric Nonlinear Functional Analysis" by Benyamini and Lindenstrauss](https://books.google.pl/books/about/Geometric_Nonlinear_Functional_Analysis.html?id=lXZ95EKwjYUC&redir_esc=y):


**Theorem (Schoenberg).** A metric space $(X,d)$ admits an isometric embedding into a Hilbert space if and only if the function $d^2$ is negative definite in the sense that $\sum d^2(x\_i,x\_j)c\_i\bar c\_j\le 0$ for all $x\_1,\dots,x\_n\in X$ and all complex scalars $c\_1,\dots,c\_n$ satisfying $\sum c\_j=0$.


This characterization has a bipartite version:


**Theorem (??).** A metric space $(X,d)$ admits an isometric embedding into a Hilbert space if and only if 
$$\sum\_{i<j} d^2(x^+\_i,x^+\_j)+\sum\_{i<j} d^2(x^-\_i,x^-\_j)\le \sum\_{i,j=1}^n d^2(x^+\_i,x\_j^-)$$ for any points $x^+\_1,\dots,x^+\_n$ and $x^-\_1,\dots,x^-\_n$ in $X$.



> 
> **Question.** I hope Theorem (??) is known. If yes, could you provide me with a suitable reference?
> 
> 
> 


The reduction of Theorem (??) to Schoenberg's Theorem can be done in six steps:


**Lemma.** For a pseudometric $d$ on a set $X$ the following conditions are equivalent:


(1) $\sum d^2(x\_i,x\_j)c\_i c\_j\le 0$ for any $x\_1,\dots,x\_n\in X$ and any complex numbers $c\_1,\dots,c\_n$ with $\sum c\_j=0$.


(2) $\sum d^2(x\_i,x\_j)c\_i c\_j\le 0$ for any $x\_1,\dots,x\_n\in X$ and any real numbers $c\_1,\dots,c\_n$ with $\sum c\_j=0$.


(3) $\sum d^2(x\_i,x\_j)c\_i c\_j\le 0$ for any $x\_1,\dots,x\_n\in X$ and any rational numbers $c\_1,\dots,c\_n$ with $\sum c\_j=0$.


(4) $\sum d^2(x\_i,x\_j)c\_i c\_j\le 0$ for any $x\_1,\dots,x\_n\in X$ and any integer numbers $c\_1,\dots,c\_n$ with $\sum c\_j=0$.


(5) $\sum d^2(x\_i,x\_j)c\_i c\_j\le 0$ for any $x\_1,\dots,x\_n\in X$ and any integer numbers $c\_1,\dots,c\_n\in\{-1,1\}$ with $\sum c\_j=0$.


(6) $\sum\_{i,j=1}^m d^2(x^+\_i,x^+\_j)+\sum\_{i,j=1}^m d^2(x^-\_i,x^-\_j)\le \sum\_{i,j=1}^m (d^2(x^+\_i,x\_j^-)+d^2(x^-\_i,x^+\_j))$ for any points $x^+\_1,\dots,x^+\_m$ and $x^-\_1,\dots,x^-\_m$ in $X$.


*Proof.* The implications $(1)\Rightarrow(2)\Rightarrow(3)\Rightarrow(4)\Rightarrow(5)$ are trivial.


$(2)\Rightarrow(1)$ Take any complex numbers $c\_1,\dots,c\_n$ with $\sum c\_k=0$. Write each complex number $c\_k$ as $c\_k=a\_k+ b\_ki$. The equality $\sum c\_k=0$ implies $\sum a\_k=0=\sum b\_k$. Consider the number $s=\sum d^2(x\_i,x\_j)c\_i\bar c\_j$ and observe that it is real: $\bar s=\sum d^2(x\_i,x\_j)\bar c\_ic\_j=\sum d^2(x\_j,x\_i)c\_j\bar c\_i=s$. Then
$$s=\sum d^2(x\_i,x\_j)c\_i\bar c\_j=\sum d^2(x\_ix\_j)(a\_ia\_j+b\_ib\_j)=\sum d^2(x\_ix\_j)a\_ia\_j+\sum d^2(x\_i,x\_j)b\_ib\_j\le 0$$by (2).


$(3)\Rightarrow(2)$ can be proved by a standard continuity argument.


$(4)\Rightarrow(3)$ can be proved by multiplying the rational numbers $c\_1,\dots,c\_n$ by their common denominator.


$(5)\Rightarrow(4)$ can be proved by repeating each point $x\_i$ $|c\_i|$ times.


$(5)\Leftrightarrow(6)$ The condition (6) is a rewritten condition (5) with $m=n/2$ and points $x^+\_1,\dots,x^+\_m$ corresponding to $c\_i=1$ and $x^-\_1,\dots,x^-\_m$ to $c\_i=-1$.



A reference to a characterization of metric spaces admitting an isometric embedding into a Hilbert space
In "[Geometry of cuts and metrics](https://www.researchgate.net/publication/242569526_Geometry_of_Cuts_and_Metrics)" by Deza and Laurent, your inequality is called *pure inequality of negative type*.


In Section 6.1.1, it is stated that pure inequalities of negative type imply all inequalities of negative type.


In Theorem 6.2.2 they formulate Schoenberg's criterion: a metric space admits an isometric embedding into a Hilbert space if and only it the square of the metric satisfies all the inequalities of negative type.


