Setup:
Let $\Gamma$ be the set of non-positively curved weighted connected graphs, with finitely many points, which are isometrically embedded in $\mathbb{R}^n$; for some $n\in \mathbb{N}$;$n\geq 2$. Thus, an element $\gamma$ of $\Gamma$ is a triple $(E_{\gamma},V_{\gamma},W_{\gamma})$ where $E_{\gamma}$ and $V_{\gamma}$ are respectively finite sets of edges and vertices and where the weights $W_{\gamma}:V_{\gamma}\rightarrow \mathbb{R}$ satisfy: there exists some $\delta>0$ such that:
$$
\min_{e\in E_{\gamma}}\, \min_{v \in V_{\gamma}(e)} W_{\gamma}(v)>\delta;
$$
where $V_{\gamma}(e)$ denotes the set of vertices in $V_{\gamma}$ which are adjacent to $e$.
We make $\gamma$ into a metric space by equipping it with the usual graph distance:
$$
d_{\gamma}(e,\tilde{e}) := \min_{[x]} \sum_{[x]} \, W_{\gamma}(x)
,
$$
where the infimum is taken over all sequences $[x]:=(e,\dots,\tilde{e})$ of vertices beginning with $e$ and terminating at $\tilde{e}$.
Background:
We note that, on page 4 (d) of Ballmann - Lectures on spaces of nonpositive curvature, any such $\gamma$ is a metric space of non-positive curvature.
This is because:
- It is geodesic,
- For any $e_1,e_2,e_3 \in \gamma$ we have: $$ d_{\gamma}(e_1,m) \leq \sqrt{2^{-1}d_{\gamma}^2(e_1,e_2)+ 2^{-1}d_{\gamma}^2(e_1,e_3) - 2^{-2}d_{\gamma}(e_2,e_3)} . $$
Question:
My question is: What is the closure of $\Gamma$ in the Gromov–Hausdorff space of embedded compact metric subspaces of $\mathbb{R}^n$? For example, does it contain all non-empty compacts subsets of $\mathbb{R}^n$?
