Independence of synthetic curvature dimension conditions on transport distance exponent

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 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 CDp(K, N) the analogous condition but with the cost as the pth power of the distance, we show that CDp(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 CDp(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 CDp(K, N) spaces.