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 
.
