Equivalent Definitions of Very Strict CD(K,N) -spaces

Abstract

We show the equivalence of the definitions of very strict CD(K,N) -condition defined, on one hand, using (only) the entropy functionals, and on the other, the full displacement convexity class mathcal{D}mathcal{C}_N. In particular, we show that assuming the convexity inequalities for the critical exponent implies it for all the greater exponents. We also establish the existence of optimal transport maps in very strict CD(K,N) -spaces with finite N.