We study in this paper the relationship of isoperimetric inequality and hyperbolicity for graphs and Riemannian manifolds. We obtain a characterization of graphs and Riemannian manifolds (with bounded local geometry) satisfying the (Cheeger) isoperimetric inequality, in terms of their Gromov boundary, improving similar results from a previous work. In particular, we prove that having a pole is a necessary condition to have isoperimetric inequality and, therefore, it can be removed as hypothesis.

中文翻译：

---

关于 Gromov 双曲流形和图的等周不等式的注记

我们在本文中研究了图和黎曼流形的等周不等式和双曲性的关系。我们根据 Gromov 边界获得了满足（Cheeger）等周不等式的图和黎曼流形（具有有界局部几何）的特征，改进了先前工作的类似结果。特别是，我们证明了有一个极点是等周不等式的必要条件，因此，它可以作为假设去除。