论文标题
尖锐而刚性的等法不平等在公制尺寸空间中，具有非负RICCI曲率的空间
Sharp and rigid isoperimetric inequality in metric measure spaces with non-negative Ricci curvature
论文作者
论文摘要
通过使用最佳运输理论，我们证明了涉及体积熵的无尺寸的等轴测不平等，在Lott-Sturm-andrm-villani的意义上，具有非负RICCI曲率的度量度量空间。我们表明，当且仅当该空间满足某个叶面特性时，当且仅当空间符合某个叶面特性时，就可以实现这种等法不平等。对于具有非负riemannian ricci曲率的度量度量空间，我们表明，当且仅当一个维一个维空间被拆分时，且仅当时是通过非平凡的可测量集来实现尖锐的脸颊常数。我们的等速度不平等和刚性定理在非平滑框架中证明，但即使在平滑环境中也是新的。特别是，我们的结果提供了对对数凹入措施的一些新理解。
By using optimal transport theory, we prove a sharp dimension-free isoperimetric inequality involving the volume entropy, in metric measure spaces with non-negative Ricci curvature in the sense of Lott--Sturm--Villani. We show that this isoperimetric inequality is attained by a non-trivial open set, if and only if the space satisfies a certain foliation property. For metric measure spaces with non-negative Riemannian Ricci curvature, we show that the sharp Cheeger constant is achieved by a non-trivial measurable set, if and only if a one-dimensional space is split off. Our isoperimetric inequality and the rigidity theorems are proved in non-smooth framework, but new even in the smooth setting. In particular, our results provide some new understanding of logarithmically concave measures.
