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:

**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) $\sum d^2(x_i,x_j)c_i \bar c_j\le 0$ 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 $c_i=-1$.

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