The Cheeger problem in abstract measure spaces

TL;DR

This work extends the Cheeger problem to highly abstract measure spaces endowed with a perimeter functional, removing the need for an ambient metric. By building BV and Sobolev spaces via a coarea formula tied to P, the authors develop existence and structural results for N-Cheeger sets, derive connections to prescribed P-mean curvature, and establish links to the first Dirichlet 1-Laplacian eigenvalue, as well as to Dirichlet p-eigenvalues and p-torsion. The framework yields general isoperimetric-type inequalities, inequalities between N- and M-Cheeger constants, and localization properties for subclusters, covering a broad spectrum of settings from Euclidean spaces with densities to nonlocal and geometric contexts like Riemannian and sub-Riemannian spaces, CD-spaces, and metric graphs. Overall, the paper provides a unifying, minimal-assumption approach to isoperimetric, spectral, and curvature-type problems in very general spaces, while recovering and extending classical results in numerous concrete settings.

Abstract

We consider non-negative $σ$-finite measure spaces coupled with a proper functional $P$ that plays the role of a perimeter. We introduce the Cheeger problem in this framework and extend many classical results on the Cheeger constant and on Cheeger sets to this setting, requiring minimal assumptions on the pair measure space-perimeter. Throughout the paper, the measure space will never be asked to be metric, at most topological, and this requires the introduction of a suitable notion of Sobolev spaces, induced by the coarea formula with the given perimeter.

Theorems & Definitions (86)

• Definition 2.1
• Remark 2.2: $P$ is invariant under $\mathfrak{m}$-negligible modifications
• Remark 2.3
• Lemma 2.4: Total variation of sets
• proof
• Remark 2.5
• Lemma 2.6: Basic properties of total variation
• proof
• Proposition 2.7: Convexity of variation
• Definition 2.8