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 ${\rm BCD}(K,N)$ is equivalent to ${\rm RCD}(K,N)$ for $K\in \mathbb R$ and $1<N<\infty$. 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 ${\rm BCD}$ 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 ${\rm BCD}$ 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.