By way of motivation:
Let $f:[0,1]\rightarrow\mathbb{R}^2$ and $g:\mathbb{S}_1\rightarrow\mathbb{R}^2$ be continuous injections, where $\mathbb{S}_1$ is the circle ($\mathbb{R}/\mathbb{Z}$).
Then, by Bochner's theorem, for any $\lambda>0$, any $n$, any $x_1,\dots,x_n\in f([0,1])$ and any $y_1,\dots,y_n \in g(\mathbb{S}_1)$, the matrices:
- $\left[\exp(-\lambda|f^{-1}(x_i)-f^{-1}(x_j)|)\right]_{i,j=1,\dots,n}$ and
- $\left[\exp(-\lambda d(g^{-1}(y_i),g^{-1}(y_j)))\right]_{i,j=1,\dots,n}$
are positive definite, where $d$ is the usual metric on $\mathbb{S}_1$.
I.e. the function $d\mapsto\exp(-\lambda d)$ is positive definite on these spaces, in the sense of Schoenberg.
My question is the following:
Let $A\subset \mathbb{R}_2$ be the embedding of a planar graph in the plane (i.e. a closed subset comprised of joined curves). Is $d\mapsto \exp(-\lambda d)$ positive definite on $A$, with the intrinsic (path) metric? Could positive definiteness of this function be preserved under the operation of "gluing"?
A sufficient condition:
It is sufficient to show that a weighted planar graph with the shortest-path metric may be isometrically embedded in $\mathbb{S}_{n_1}\times \mathbb{S}_{n_2} \times \dots \times \mathbb{S}_{n_p}$, with the weighted L1 type metric $d(s_1,s_2)=\sum_{i=1}^p{\lambda_i d_{n_i}(s_{1,i},s_{2,i})}$, where $d_n$ is the usual metric on $\mathbb{S}_n$, and where $\lambda_1,\dots,\lambda_p>0$.