Displacement Convexity and Ricci Curvature

Definitions and mathematical statement

Let $(M,g)$ be the smooth Riemannian manifold intended in the source, with Riemannian volume $\mathrm{vol}_g$. In the standard complete connected boundaryless setting, let $\mathcal P_2(M)$ be the Borel probability measures of finite second moment, equipped with \[ W_2(\mu,\nu)^2=\inf_{\pi\in\Pi(\mu,\nu)}\int d_g(x,y)^2\,{\,\mathrm{d}}\pi(x,y). \] Use the convex-sign entropy \[ \operatorname{Ent}(\mu\mid\mathrm{vol}_g)=\int_M\rho\log\rho\,{\,\mathrm{d}}\mathrm{vol}_g \quad\text{when }\mu=\rho\,\mathrm{vol}_g, \] with appropriate extended-value and integrability conventions. The opposite Shannon sign $-\int\rho\log\rho$ would reverse convexity, so that convention is not silently substituted.

For finite-entropy endpoints and a constant-speed optimal-transport geodesic $(\mu_t)_{0\leq t\leq1}$, $K$-displacement convexity means the inequality \[ \operatorname{Ent}(\mu_t\mid\mathrm{vol}_g) \leq(1-t)\operatorname{Ent}(\mu_0\mid\mathrm{vol}_g) +t\operatorname{Ent}(\mu_1\mid\mathrm{vol}_g) -\frac K2t(1-t)W_2(\mu_0,\mu_1)^2. \] The source asks whether this property is equivalent to \[ \operatorname{Ric}_g(v,v)\geq K\,g(v,v)\qquad(v\in TM), \] with $K=0$ giving the first nonnegative-Ricci question. The bound is on the unweighted Ricci tensor and the reference is Riemannian volume, not an unspecified weighted measure or different entropy functional.

The source does not state completeness, boundary conditions, permissible endpoint measures, noncompact integrability conventions, or whether the inequality is required along every eligible geodesic or along an appropriate geodesic for each endpoint pair. These choices must be fixed to identify its exact equivalence theorem. Restriction to regular compactly supported test measures or a global finite-entropy statement should not be silently exchanged without the relevant extension argument.

The supplied literature triage checked 2026-08-21 labels the extracted question open and reports no authoritative complete resolution found, but provides no separate theorem or precise missing hypothesis. That label does not certify that the familiar smooth unweighted characterization is a new unresolved claim; source review must distinguish an explanatory characterization question from any intended unresolved extension.

Short English statement

Relate lower Ricci curvature to the exact Wasserstein entropy-convexity inequality, with the $-K t(1-t)W_2^2/2$ correction. Entropy sign, reference measure, manifold hypotheses, and geodesic quantifiers must be fixed before assessing the source's open label.

Sources

• Local UnsolvedMath registry, permanent ID 30000621, catalogue entry, Displacement Convexity and Ricci Curvature.
• Classical and Quantum Mechanical Models of Many-Particle Systems (2007), Oberwolfach Reports, pp. 3189--3258, DOI https://doi.org/10.4171/owr/2006/54.
• Supplied source-verification claim and literature triage checked 2026-08-21 cite only the report; no separate equivalence theorem, entropy convention, or unresolved-extension formulation is supplied.