Wasserstein Barycenter Convexity Detects Hilbertian Geometry
We prove that convexity of the Boltzmann entropy at Wasserstein barycenters is strong enough to distinguish Hilbert spaces from general Banach spaces. Thus Wasserstein barycenters provide an intrinsic optimal-transport test for Hilbertian geometry. More precisely, we show that if a finite-dimensional normed vector space, equipped with Lebesgue measure, satisfies the Wasserstein Jensen's inequality for the entropy at barycenters of arbitrary finite families of probability measures, then its norm must be induced by an inner product. This contrasts sharply with a well-known result: every finite-dimensional normed vector space satisfies the nonnegative Ricci curvature condition in the sense of Lott--Sturm--Villani, whereas barycenter convexity excludes all non-Hilbertian norms. As a consequence, smooth reversible Finsler manifolds satisfying the corresponding barycentric curvature-dimension condition have Riemannian tangent norms. The proof does not assume strict convexity of the norm. Its two main ingredients are a rank-one polarization argument, which yields the dual parallelogram identity in the strictly convex case, and a maximal-face trapping argument, which rules out flat faces of the unit ball.