*Preprint*

**Inserted:** 29 sep 2024

**Last Updated:** 29 sep 2024

**Year:** 2024

**Notes:**

This version is new and `needle decomposition free'.

**Abstract:**

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, totally dimension-free, new even in the smooth setting. In particular, our results provide some new understanding of logarithmically concave measures.

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