PreprintCC BY 4.0arXiv
Metric foundations of geometry
Abstract
A metric space is called all-set-homogeneous if every isometry between two of its subsets extends to an isometry of the whole space. We classify all-set-homogeneous geodesic spaces: besides the classical examples, they include the universal metric trees of finite valence. We also prove that every complete all-set-homogeneous length space is geodesic, and hence the same classification holds in this setting.
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BibTeX
@misc{lebedeva2026metric,
title = {{Metric foundations of geometry}},
author = {Nina Lebedeva and Anton Petrunin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26410},
eprint = {2609.26410},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26410},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Metric foundations of geometry
AU - Nina Lebedeva
AU - Anton Petrunin
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.26410
UR - https://arxiv.org/abs/2609.26410
AB - A metric space is called all-set-homogeneous if every isometry between two of its subsets extends to an isometry of the whole space. We classify all-set-homogeneous geodesic spaces: besides the classical examples, they include the universal metric trees of finite valence. We also prove that every complete all-set-homogeneous length space is geodesic, and hence the same classification holds in this setting.
ER -