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In this note we show that in metric measure spaces satisfying the reducedcurvature-dimension condition CD*(K,N) we always have geodesics in theWasserstein space of probability measures that satisfy the critical convexityinequality of CD*(K,N) also for intermediate times and in addition the measuresalong these geodesics have an upper-bound on their densities. This upper-bounddepends on the bounds for the densities of the end-point measures, thelower-bound K for the Ricci-curvature, the upper-bound N for the dimension, andon the diameter of the union of the supports of the end-point measures.

Improved geodesics for the reduced curvature-dimension condition in branching metric spaces