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PreprintCC BY 4.0arXiv

Hausdorff dimension of the singular set in optimal transport

  • Shibing Chen
  • Qi-Rui Li
  • Yuanyuan Li
  • Jiakun Liu
  • Xiaotian Wu
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Abstract

We study optimal transport between bounded smooth domains in ℝn, with smooth positive densities bounded above and below. Without convexity assumptions on the domains, we prove that the optimal map is smooth outside a relatively closed set of Hausdorff dimension at most n−1.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026hausdorff,
  title = {{Hausdorff dimension of the singular set in optimal transport}},
  author = {Shibing Chen and Qi-Rui Li and Yuanyuan Li and Jiakun Liu and Xiaotian Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28228},
  eprint = {2610.28228},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28228},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Hausdorff dimension of the singular set in optimal transport
AU  - Shibing Chen
AU  - Qi-Rui Li
AU  - Yuanyuan Li
AU  - Jiakun Liu
AU  - Xiaotian Wu
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28228
UR  - https://arxiv.org/abs/2610.28228
AB  - We study optimal transport between bounded smooth domains in ℝ^n, with smooth positive densities bounded above and below. Without convexity assumptions on the domains, we prove that the optimal map is smooth outside a relatively closed set of Hausdorff dimension at most n−1.
ER  -