Independence of synthetic curvature dimension conditions on transport distance exponent

Abstract:

The celebrated Lott-Sturm-Villani theory of metric measure spaces furnishes synthetic notions of a Ricci curvature lower bound $K$ joint with an upper bound $N$ on the dimension. Their condition, called the Curvature-Dimension condition and denoted by $\mathsf {CD}(K,N)$, is formulated in terms of a modified displacement convexity of an entropy functional along $W_{2}$-Wasserstein geodesics. We show that the choice of the squared-distance function as transport cost does not influence the theory. By denoting with $\mathsf {CD}_{p}(K,N)$ the analogous condition but with the cost as the $p^{th}$ power of the distance, we show that $\mathsf {CD}_{p}(K,N)$ are all equivalent conditions for any $p>1$ — at least in spaces whose geodesics do not branch.

Following Cavalletti and Milman [The Globalization Theorem for the Curvature Dimension Condition, preprint, arXiv:1612.07623], we show that the trait d’union between all the seemingly unrelated $\mathsf {CD}_{p}(K,N)$ conditions is the needle decomposition or localization technique associated to the $L^{1}$-optimal transport problem. We also establish the local-to-global property of $\mathsf {CD}_{p}(K,N)$ spaces.