A topological characterization of indecomposable sets of finite perimeter
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arXiv
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| Format: | Preprint |
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2025
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| author | Bonicatto, Paolo Lahti, Panu Pasqualetto, Enrico |
| author_facet | Bonicatto, Paolo Lahti, Panu Pasqualetto, Enrico |
| contents | We prove that a set of finite perimeter is indecomposable if and only if it is, up to a choice of suitable representative, connected in the 1-fine topology. This gives a topological characterization of indecomposability which is new even in Euclidean spaces. Our approach relies crucially on the metric space theory of functions of bounded variation, and we are able to prove our main result in a complete, doubling metric measure space supporting a $1$-Poincaré inequality and having the two-sidedness property (this class includes all Riemannian manifolds, Carnot groups, and ${\sf RCD}(K,N)$ spaces with $K\in\mathbb R$ and $N<\infty$). As an immediate corollary, we obtain an alternative proof of the decomposition theorem for sets of finite perimeter into maximal indecomposable components. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_18319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A topological characterization of indecomposable sets of finite perimeter Bonicatto, Paolo Lahti, Panu Pasqualetto, Enrico Metric Geometry Functional Analysis 30L99, 26B30, 46E36 We prove that a set of finite perimeter is indecomposable if and only if it is, up to a choice of suitable representative, connected in the 1-fine topology. This gives a topological characterization of indecomposability which is new even in Euclidean spaces. Our approach relies crucially on the metric space theory of functions of bounded variation, and we are able to prove our main result in a complete, doubling metric measure space supporting a $1$-Poincaré inequality and having the two-sidedness property (this class includes all Riemannian manifolds, Carnot groups, and ${\sf RCD}(K,N)$ spaces with $K\in\mathbb R$ and $N<\infty$). As an immediate corollary, we obtain an alternative proof of the decomposition theorem for sets of finite perimeter into maximal indecomposable components. |
| title | A topological characterization of indecomposable sets of finite perimeter |
| topic | Metric Geometry Functional Analysis 30L99, 26B30, 46E36 |
| url | https://arxiv.org/abs/2512.18319 |