Preprint
Inserted: 5 jul 2026
Last Updated: 5 jul 2026
Year: 2026
Abstract:
A striking feature of the Lott--Sturm--Villani curvature-dimension condition is its Finsler flatness phenomenon: every finite-dimensional normed space equipped with Lebesgue measure satisfies the weak $\mathrm {CD}(0,n)$ condition, independently of whether the norm is Euclidean. This phenomenon, emphasized by Villani as both natural and surprising, shows that entropy convexity along Wasserstein geodesics alone does not distinguish Hilbertian flat geometry from genuinely non-Hilbertian Finsler geometry.
We show that Wasserstein barycentric convexity provides precisely such a distinction. We prove that if a finite-dimensional normed vector space, equipped with Lebesgue measure, satisfies the Wasserstein Jensen's inequality for the Boltzmann entropy at barycenters, then its norm must be induced by an inner product. Thus Wasserstein barycenters provide an intrinsic optimal-transport test for Hilbertian structures. As a consequence, smooth reversible Finsler manifolds satisfying the corresponding barycentric curvature-dimension condition have Riemannian tangent norms.
The proof does not assume smoothness or strict convexity of the norm. Its two main ingredients, a rank-one polarization argument and a maximal-face trapping argument, are also of independent interest for the optimal transport theory.
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