Steiner Minimal Trees in the Gromov--Hausdorff Space and in Gromov--Hausdorff Clouds

We prove the existence of Steiner minimal trees spanning a finite family of metric spaces in two cases of the Gromov–Hausdorff theory. In the compact Gromov–Hausdorff space $\mathcal{M}$, every finite family of nonempty compact metric spaces is spanned by a shortest Steiner minimal tree whose Steiner vertices are again nonempty compact metric spaces. In a fixed Gromov–Hausdorff cloud $\mathcal{C}$, every finite family of proper, isometrically cobounded spaces admits a finite-length Steiner minimal tree all of whose vertices remain in $\mathcal{C}$. The compact case uses compact replacement through distance profiles and the Hausdorff hyperspace, and the cloud case uses pointed ultralimits.