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

DOI10.1007/S12220-022-01068-XarXiv1906.07693MaRDI QIDQ6320702

Publication date: 18 June 2019

Abstract: We show the equivalence of the definitions of very strict

C D (K, N)

-condition defined, on one hand, using (only) the entropy functionals, and on the other, the full displacement convexity class

m a t h c a l {D 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

C D (K, N)

-spaces with finite

N

.

Mathematics Subject Classification ID

Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces (53C23) Optimal transportation (49Q22)

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