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Gromov-compactifiability of Wasserstein spaces
Abstract
We characterize Gromov-compactifiability of Wasserstein spaces over Polish metric spaces for 1β€ pβ€β. The space π«1(X) is Gromov-compactifiable if and only if X is bounded and Gromov-compactifiable, whereas π«p(X) is Gromov-compactifiable for every 1<p<β. At the other endpoint, π«β(X) is Gromov-compactifiable if and only if X is. For p=1, the sufficiency argument combines an averaged triangle defect with tightness. For 1<p<β, the main tool is the uniform convexity of Lp. The p=β proof uses the finite-set criterion and a mass redistribution construction.
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BibTeX
@misc{jiang2026gromovcompactifiability,
title = {{Gromov-compactifiability of Wasserstein spaces}},
author = {Huajian Jiang and Zhi-Xiang Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12355},
eprint = {2609.12355},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12355},
}
RIS
TY - UNPB
TI - Gromov-compactifiability of Wasserstein spaces
AU - Huajian Jiang
AU - Zhi-Xiang Zhu
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12355
UR - https://arxiv.org/abs/2609.12355
AB - We characterize Gromov-compactifiability of Wasserstein spaces over Polish metric spaces for 1β€ pβ€β. The space π«_1(X) is Gromov-compactifiable if and only if X is bounded and Gromov-compactifiable, whereas π«_p(X) is Gromov-compactifiable for every 1<p<β. At the other endpoint, π«_β(X) is Gromov-compactifiable if and only if X is. For p=1, the sufficiency argument combines an averaged triangle defect with tightness. For 1<p<β, the main tool is the uniform convexity of L^p. The p=β proof uses the finite-set criterion and a mass redistribution construction.
ER -