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We construct geodesics in the Wasserstein space of probability measure alongwhich all the measures have an upper bound on their density that is determinedby the densities of the endpoints of the geodesic. Using these geodesics weshow that a local Poincar\'e inequality and the measure contraction propertyfollow from the Ricci curvature bounds defined by Sturm. We also show for alarge class of convex functionals that a local Poincar\'e inequality is impliedby the weak displacement convexity of the functional.

Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm