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We show the equivalence of the definitions of very strict $CD(K,N)$-condition defined, on one hand, using (only) the entropy functionals, and onthe other, the full displacement convexity class $\mathcal{DC}_N$. Inparticular, we show that assuming the convexity inequalities for the criticalexponent implies it for all the greater exponents. We also establish theexistence of optimal transport maps in very strict $CD(K,N)$ -spaces withfinite $N$.

Equivalent definitions of very strict $CD(K,N)$ -spaces