In this paper we study the Cheeger cut and Cheeger problem in the general framework of metric measure spaces. A central motivation for developing our results has been the desire to unify the assumptions and methods employed in various specific spaces, such as Riemannian manifolds, Heisenberg groups, graphs, etc. We obtain two characterization of the Cheeger constant: a variational one and another one through the eigenvalue of the 1-Laplacian. We obtain a Cheeger inequality along the lines of the classical one for Riemannian manifolds obtained by Cheeger in (In: Gunning RC (ed) Problems in analysis. Princeton University Press, Princeton, pp 195–199, 1970). We also study the Cheeger problem. Through a variational characterization of the Cheeger sets we prove the existence of Cheeger sets and obtain a characterization of the calibrable sets and a version of the Max Flow Min Cut Theorem.

本文在度量测度空间的一般框架下研究了 Cheeger 割和 Cheeger 问题。开发我们的结果的一个核心动机是希望统一各种特定空间中使用的假设和方法，例如黎曼流形、海森堡群、图等。我们获得了 Cheeger 常数的两种表征：一种是变分的，另一种是变分的通过 1-拉普拉斯算子的特征值。我们沿着 Cheeger 在（In：Gunning RC (ed) Problems in Analysis. Princeton University Press, Princeton, pp 195–199, 1970）中获得的黎曼流形的经典不等式获得了 Cheeger 不等式。我们还研究了奇格问题。通过 Cheeger 集的变分表征，我们证明了 Cheeger 集的存在性，并获得了可校准集的表征和最大流最小割定理的版本。