A reference to a characterization of metric spaces admitting an isometric embedding into a Hilbert space

Question

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

Answer 1

In "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.

Answer 2

This nice observation is new to me. In particular, I do not know a reference. Thus this is not an answer, rather a comment which I want to highlight. My comment is that you can supplement your observation with the following one, which could be proven similarly, based on an analogous reduction of positive definite kernels.

Claim: A $[0,\pi]$-valued metric space $(X,d)$ admits an isometric embedding into a round sphere (possibly infinite dimensional) iff for every naturals $m,n$ and elements $x^+_1,\ldots x^+_n,x^-_1,\ldots x^-_m \in X$ we have $$ \frac{1}{2}(n+m)+ \sum_{i<j\leq n} \cos(d(x^+_i,x^+_j)) + \sum_{i<j\leq m} \cos(d(x^-_i,x^-_j)) \geq \sum_{i\leq n,j\leq m} \cos(d(x^+_i,x^-_j)).$$

I suppose also that upon replacing $\cos$ with $\cosh$ you can have a similar iff criterion for embedablility in hyperbolic spaces, but I didn't check the details here.