熵的位移凸性与距离费用最优运输
Displacement convexity of Entropy and the distance cost Optimal Transportation
摘要：
在过去的十年中，最优输运在奇异空间的几何研究中发挥了重要作用，并最终提出了洛特-斯特姆-维拉尼理论。后者是建立在里奇曲率下界的表征某些熵泛函沿$W_{2}$ -测地线的位移凸性。考虑到$L^1$的不同观点，理论（局部化范式和局部到全局性质）的实质性最新进展已经获得-最优输运问题产生不同的曲率维度$\mathsf{CD}^{1}(K,N)$ [8]，根据Lipschitz映射的积分曲线的一维曲率性质来制定。在本说明中，我们表明，这两种方法产生相同的曲率维数条件，调和两个定义。特别是，我们表明，$\mathsf{CD}^{1}(K,N)$条件可以制定在沿$W_{1}$ -测地线位移凸。
英文摘要：
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.
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