The Cheeger problem in abstract measure spaces

Abstract

We consider non-negative $\ddot{I}ƒ$-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.

52 pages - There is a minor mistake in the proof of Theorem 3.6 in the published version: when estimating P(E_k(i)) from above, one needs to bound it with m(Om)(h_N(Om)+1) rather than with 2m(Om)h_N(Om) (as h_N(Om) might be zero). The following inequalities change accordingly. The preprint contains the amended statement