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Mathematics > Metric Geometry

arXiv:2207.00482 (math)
[Submitted on 1 Jul 2022 (v1), last revised 26 Dec 2023 (this version, v2)]

Title:The Cheeger problem in abstract measure spaces

Authors:Valentina Franceschi, Andrea Pinamonti, Giorgio Saracco, Giorgio Stefani
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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.
Comments: 52 pages
Subjects: Metric Geometry (math.MG); Functional Analysis (math.FA)
MSC classes: Primary 49Q20, Secondary 35P15, 53A10
Cite as: arXiv:2207.00482 [math.MG]
  (or arXiv:2207.00482v2 [math.MG] for this version)
  https://doi.org/10.48550/arXiv.2207.00482
Journal reference: J. London Math. Soc. 119(1):e12840, 2024
Related DOI: https://doi.org/10.1112/jlms.12840

Submission history

From: Giorgio Saracco [view email]
[v1] Fri, 1 Jul 2022 15:14:19 UTC (61 KB)
[v2] Tue, 26 Dec 2023 19:33:30 UTC (53 KB)
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