Set-decomposition of normal rectifiable G-chains via an abstract decomposition principle
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Michael Goldman [1] ; Benoit Merlet [2]
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[1] Institut Polytechnique de Paris, Palaiseau, France
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[2] EPE Université de Lille, Lille, France
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- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 40, Nº 6, 2024, págs. 2073-2094
- Idioma: inglés
- DOI: 10.4171/RMI/1504
- Enlaces
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Resumen
We introduce the notion of set-decomposition of a normal G-flat chain A in Rn as a sequence A j=A└S j associated to a Borel partition Sj of Rn such that N(A)=∑N(Aj). We show that any normal rectifiable G-flat chain admits a decomposition in set-indecomposable sub-chains. This generalizes the decomposition of sets of finite perimeter in their “measure theoretic” connected components due to Ambrosio, Caselles, Masnou and Morel. It can also be seen as a variant of the decomposition of integral currents in indecomposable components by Federer. As opposed to previous results, we do not assume that G is boundedly compact. Therefore, we cannot rely on the compactness of sequences of chains with uniformly bounded N-norms. We deduce instead the result from a new abstract decomposition principle. As in earlier proofs, a central ingredient is the validity of an isoperimetric inequality. We obtain it here using the finiteness of some h-mass to replace integrality.
