On the Geometry of Wasserstein Barycenter II: Riemannian Rigidity, Essential Non-Branching, and Finsler Models
Abstract
Wasserstein barycenters provide a notion of weighted mean for probability measures on a metric space. We prove that the barycenter curvature-dimension condition BCD(K,N) is equivalent to RCD(K,N) for K∈ ℝ and 1<N<∞. This gives a new characterization of Riemannian curvature-dimension spaces by entropy inequalities at barycenters of finite families of measures. As a byproduct, we introduce an almost BCD condition that allows an additive error in the entropy inequality. It is stable under measured Gromov-Hausdorff convergence and implies essential non-branching when the error is sufficiently small. The resulting class contains non-Riemannian Finsler spaces. Within the BCD framework, this answers an open problem posed by Ambrosio in his 2018 ICM survey. We further bound the failure of the parallelogram identity for cotangent norms in terms of the entropy error.
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BibTeX
@misc{han2026geometry,
title = {{On the Geometry of Wasserstein Barycenter II: Riemannian Rigidity, Essential Non-Branching, and Finsler Models}},
author = {Bang-Xian Han and Deng-Yu Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.19564},
eprint = {2609.19564},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19564},
}
RIS
TY - UNPB
TI - On the Geometry of Wasserstein Barycenter II: Riemannian Rigidity, Essential Non-Branching, and Finsler Models
AU - Bang-Xian Han
AU - Deng-Yu Liu
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2609.19564
UR - https://arxiv.org/abs/2609.19564
AB - Wasserstein barycenters provide a notion of weighted mean for probability measures on a metric space. We prove that the barycenter curvature-dimension condition BCD(K,N) is equivalent to RCD(K,N) for K∈ ℝ and 1<N<∞. This gives a new characterization of Riemannian curvature-dimension spaces by entropy inequalities at barycenters of finite families of measures. As a byproduct, we introduce an almost BCD condition that allows an additive error in the entropy inequality. It is stable under measured Gromov-Hausdorff convergence and implies essential non-branching when the error is sufficiently small. The resulting class contains non-Riemannian Finsler spaces. Within the BCD framework, this answers an open problem posed by Ambrosio in his 2018 ICM survey. We further bound the failure of the parallelogram identity for cotangent norms in terms of the entropy error.
ER -