论文标题
熵的位移凸和距离成本最佳运输
Displacement convexity of Entropy and the distance cost Optimal Transportation
论文作者
论文摘要
在过去的十年中，最佳运输在以Lott-Sturm-Villani理论为导致的奇异空间的几何形状中具有相关作用。后者建立在RICCI曲率下限的表征上，以某些熵函数的位移凸面沿$ W_ {2} $ - 大地测量学。考虑到$ l^1 $ - 最佳的运输问题的不同观点，该理论中的最新进展（本地化范式和局部到全球特性）得到了不同的曲率$ \ MATHSF {CD}^{1}^{1}（k，k，n）$的不同曲率尺寸的观点。在本说明中，我们表明两种方法产生了相同的曲率维度条件来核对这两个定义。特别是我们表明，可以根据$ W_ {1} $ - GeoDesics的位移凸面来制定$ \ Mathsf {CD}^{1}^{1}（K，N）$条件。
During the last decade Optimal Transport had a relevant role in the study of geometry of singular spaces that culminated with the Lott-Sturm-Villani theory. The latter is built on the characterisation of Ricci curvature lower bounds in terms of displacement convexity of certain entropy functionals along $W_{2}$-geodesics. Substantial recent advancements in the theory (localization paradigm and local-to-global property) have been obtained considering the different point of view of $L^1$-Optimal transport problems yielding a different curvature dimension $\mathsf{CD}^{1}(K,N)$ [8] formulated in terms of one-dimensional curvature properties of integral curves of Lipschitz maps. In this note we show that the two approaches produce the same curvature-dimension condition reconciling the two definitions. In particular we show that the $\mathsf{CD}^{1}(K,N)$ condition can be formulated in terms of displacement convexity along $W_{1}$-geodesics.
