Curvature-Dimension Condition

The curvature-dimension condition is a synthetic lower Ricci curvature bound and upper dimension bound for a metric measure space, expressed by entropy inequalities along optimal transport geodesics. For and finite , one uses the power entropy functional . For any two absolutely continuous probability measures with bounded support, there must be a Wasserstein geodesic satisfying

for and every , with the usual extension of the entropy to singular measures. For nonzero , the coefficients are replaced by the distance-dependent comparison coefficients from the constant-curvature model. On a smooth -dimensional Riemannian manifold with its volume measure, the condition for is equivalent to . The unreduced condition here is distinguished from the variant . Combining it with quadratic Cheeger energy defines .