The Cheeger problem in abstract measure spaces

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.

On the monotonicity of weighted perimeters of convex bodies

“…Clearly, if 𝑓 ≡ 𝑐 for some 𝑐 ∈ [0, +∞), then 𝑃 𝑓 = 𝑐 𝑃, a constant multiple of the Euclidean perimeter. Weighted perimeters have been largely investigated in relation to isoperimetric, cluster, and Cheeger problems, see [2,10,11,13,[17][18][19][28][29][30] and the survey [27] for an account on the existing literature.…”