Independence of synthetic Curvature Dimension conditions on transport distance exponent

TL;DR

This paper demonstrates that synthetic Ricci curvature conditions based on optimal transport are independent of the transport distance exponent for p>1, unifying various conditions through localization techniques.

Contribution

It proves the equivalence of $ ext{CD}_p(K,N)$ conditions for all p>1 in non-branching spaces and shows their independence from the choice of transport cost exponent.

Findings

$ ext{CD}_p(K,N)$ conditions are equivalent for all p>1 in non-branching spaces.

The choice of transport cost exponent does not affect the curvature-dimension condition.

Localization techniques unify different $ ext{CD}_p(K,N)$ conditions.

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 $\mathrm{CD}(K,N)$, 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 $\mathrm{CD}_{p}(K,N)$ the analogous condition but with the cost as the $p^{th}$ power of the distance, we show that $\mathrm{CD}_{p}(K,N)$ are all equivalent conditions for any $p>1$ -- at least in spaces whose geodesics do not branch. We show that the trait d'union between all the seemingly unrelated $\mathrm{CD}_{p}(K,N)$ 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 $\mathrm{CD}_{p}(K,N)$ spaces.