The Cheeger problem in abstract measure spaces

TL;DR

This paper extends the classical Cheeger problem to abstract measure spaces with a perimeter functional, establishing fundamental results without requiring metric structures, and introduces Sobolev spaces suited for this setting.

Contribution

It generalizes the Cheeger problem to non-metric measure spaces, defining Sobolev spaces via coarea formula and minimal assumptions on measure-perimeter pairs.

Findings

Extension of Cheeger constant and sets to abstract measure spaces

Introduction of Sobolev spaces in non-metric measure spaces

Minimal assumptions on measure space and perimeter

Abstract

We consider non-negative $\sigma$-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.