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The celebrated Lott-Sturm-Villani theory of metric measure spaces furnishessynthetic notions of a Ricci curvature lower bound $K$ joint with an upperbound $N$ on the dimension. Their condition, called the Curvature-Dimensioncondition and denoted by $\mathrm{CD}(K,N)$, is formulated in terms of amodified displacement convexity of an entropy functional along$W_{2}$-Wasserstein geodesics. We show that the choice of the squared-distancefunction 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 allequivalent conditions for any $p>1$ -- at least in spaces whose geodesics donot branch. We show that the trait d'union between all the seemingly unrelated$\mathrm{CD}_{p}(K,N)$ conditions is the needle decomposition or localizationtechnique associated to the $L^{1}$-optimal transport problem. We alsoestablish the local-to-global property of $\mathrm{CD}_{p}(K,N)$ spaces.

Independence of synthetic Curvature Dimension conditions on transport distance exponent