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

TL;DR

This paper proves the equivalence of different definitions of very strict $CD(K,N)$-spaces, linking entropy-based and displacement convexity approaches, and establishes the existence of optimal transport maps in these spaces.

Contribution

It demonstrates the equivalence of entropy and displacement convexity definitions of very strict $CD(K,N)$-spaces and proves the existence of optimal transport maps in finite $N$ cases.

Findings

Equivalence of entropy and displacement convexity definitions.

Convexity inequalities at critical exponents imply inequalities at all greater exponents.

Existence of optimal transport maps in very strict $CD(K,N)$-spaces with finite $N$.

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 $\mathcal{DC}_N$. 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 $N$.