论文标题
传输距离指数的合成曲率维度条件的独立性
Independence of synthetic Curvature Dimension conditions on transport distance exponent
论文作者
论文摘要
著名的Lott-Sturm-Villani公制度量空间理论提供了RICCI曲率下限$ K $关节的合成概念，其尺寸上有上限$ n $。它们的病情称为曲率维度条件，并用$ \ mathrm {cd}（k，n）$表示，是根据修改的位移凸度沿$ W_ {2} $ - Wasserstein Geodesesics配制的。我们表明，作为运输成本的平方距离功能的选择不会影响理论。通过用$ \ mathrm {cd} _ {p}（k，n）$表示类似条件，但成本为$ p^{th} $距离的功率，我们表明$ \ m arm {cd} _ {cd} _ {p} _ {p}（k，k，k，n）$都是$ p> 1 $ - 至少在spece中的所有等效条件。我们表明，所有看似无关的$ \ mathrm {cd} _ {p}（k，n）$条件之间的特质d'Union是与$ l^{1} $相关的针分解或本地化技术 - 最佳运输问题。我们还建立了$ \ mathrm {cd} _ {p}（k，n）$ spaces的本地属性属性。
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.
