  In this note we show that in metric measure spaces satisfying the reduced
curvature-dimension condition CD*(K,N) we always have geodesics in the
Wasserstein space of probability measures that satisfy the critical convexity
inequality of CD*(K,N) also for intermediate times and in addition the measures
along these geodesics have an upper-bound on their densities. This upper-bound
depends on the bounds for the densities of the end-point measures, the
lower-bound K for the Ricci-curvature, the upper-bound N for the dimension, and
on the diameter of the union of the supports of the end-point measures.
