Preprint

Inserted: 8 dec 2022
Last Updated: 8 dec 2022

Year: 2022

Abstract:

We introduce the notion of set-decomposition of a normal $G$-flat chain. 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 $\mathbb{N}$-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.

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• SetDecompositionOfGChains.pdf