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.

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中文翻译：

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合成曲率维数条件对传输距离指数的独立性

摘要：著名的 Lott-Sturm-Villani 度量空间理论提供了 Ricci 曲率下界 $K$ 联合与维度上的上限 $N$ 的综合概念。他们的条件称为曲率维条件，用 $\mathsf {CD}(K,N)$ 表示，是根据熵函数沿 $W_{2}$-Wasserstein 测地线的修正位移凸性制定的。我们表明，选择平方距离函数作为运输成本不会影响理论。通过用 $\mathsf {CD}_{p}(K,N)$ 表示类似的条件，但将成本作为 $p^{th}$ 距离的幂，我们证明 $\mathsf {CD}_ {p}(K,N)$ 都是任何 $p>1$ 的等效条件——至少在测地线不分支的空间中。遵循 Cavalletti 和 Milman [The Globalization Theorem for the Curvature Dimension Condition, preprint, arXiv:1612.07623]，我们证明了所有看似无关的 $\mathsf {CD}_{p}(K,N)$ 之间的 trait d'union条件是与$L^{1}$-最优传输问题相关的针分解或定位技术。我们还建立了 $\mathsf {CD}_{p}(K,N)$ 空间的局部到全局属性。

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