PreprintCC BY 4.0arXiv
Steiner Minimal Trees in the Gromov--Hausdorff Space and in Gromov--Hausdorff Clouds
Abstract
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 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 𝒞, every finite family of proper, isometrically cobounded spaces admits a finite-length Steiner minimal tree all of whose vertices remain in 𝒞. The compact case uses compact replacement through distance profiles and the Hausdorff hyperspace, and the cloud case uses pointed ultralimits.
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BibTeX
@misc{mikhailov2026steiner,
title = {{Steiner Minimal Trees in the Gromov--Hausdorff Space and in Gromov--Hausdorff Clouds}},
author = {I. N. Mikhailov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17400},
eprint = {2610.17400},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17400},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Steiner Minimal Trees in the Gromov--Hausdorff Space and in Gromov--Hausdorff Clouds
AU - I. N. Mikhailov
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17400
UR - https://arxiv.org/abs/2610.17400
AB - 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 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 𝒞, every finite family of proper, isometrically cobounded spaces admits a finite-length Steiner minimal tree all of whose vertices remain in 𝒞. The compact case uses compact replacement through distance profiles and the Hausdorff hyperspace, and the cloud case uses pointed ultralimits.
ER -