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  "body": "\n## Summary\n\nThe flat norm measures the distance between two [[currents|360H7MD9RD]] by the least total size of a decomposition.\nThe difference is split into a small piece plus the boundary of a small filling.\nTwo surfaces are close in the flat norm when they differ by a small area or bound a thin region.\nThis holds even if their masses differ.\nConvergence in the flat norm is the natural convergence of generalised surfaces, and the [[Federer–Fleming compactness theorem|TPXYARH912]] is stated with it.\nIt also gives the distance used in the comparison and approximation of [[integral currents|EG4AJVEFT2]].\n\n## Explanations\n\n### L1 human\n\nTwo nearby points, one counted positively and one negatively, look very different to the mass, which counts each point once and gives a total of $2$.\nBut they are the two ends of a short segment, so they are almost the same once one allows filling in.\nThe flat norm captures this: the difference is the boundary of a segment of length $\\varepsilon$, so its flat norm is at most $\\varepsilon$.\nIn the same way, a wiggly curve close to a straight one is near it in the flat norm, because the two curves bound a thin region.\n",
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  "name": "Flat norm",
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