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  "body": "\n## Definition\n\nLet $0 \\le m \\le n$, and for a [[current|360H7MD9RD]] $R$ on $\\R^n$ let $\\mathbf{M}(R) \\in [0, \\infty]$ be its mass.\nThe *flat norm* of an $m$-current $T$ on $\\R^n$ is\n$$\\mathbf{F}(T) = \\inf \\set{ \\mathbf{M}(R) + \\mathbf{M}(S) : R \\text{ an } m\\text{-current}, \\ S \\text{ an } (m+1)\\text{-current}, \\ T = R + \\partial S },$$\nwith $\\inf \\emptyset = \\infty$.\nThe *flat distance* between $m$-currents $T_1$ and $T_2$ is $\\mathbf{F}(T_1 - T_2)$.\nFor [[integral currents|EG4AJVEFT2]], the *integral flat norm* is defined in the same way with $R$ and $S$ required to be integral currents.\n\n## Remarks\n\nTaking $S = 0$ shows $\\mathbf{F}(T) \\le \\mathbf{M}(T)$.\nIf $\\mathbf{F}(T_j - T) \\to 0$, then $T_j(\\omega) \\to T(\\omega)$ for every smooth compactly supported $m$-form $\\omega$, so flat convergence implies weak convergence.\nWhen $\\mathbf{F}(T) \u003c \\infty$, there is a dual description of $\\mathbf{F}(T)$ as the supremum of $T(\\omega)$ over the forms $\\omega$ such that $\\omega$ and $d\\omega$ have comass at most $1$ everywhere.\n\nExample: in $\\R$, let $a \u003c b$ and $T = \\delta_b - \\delta_a$, a $0$-current.\nThen $T = \\partial [\\![a, b]\\!]$, so $\\mathbf{F}(T) \\le b - a$, and $\\mathbf{F}(T) \\le \\mathbf{M}(T) = 2$.\nTesting $T$ on a function $\\varphi$ with $\\abs{\\varphi} \\le 1$ and $\\abs{\\varphi'} \\le 1$, chosen with $\\varphi(b) - \\varphi(a) = \\min(b - a, 2)$, shows that $\\mathbf{F}(T) = \\min(b - a, 2)$.\nSo $\\delta_b - \\delta_a \\to 0$ in the flat norm as $b \\to a$, while its mass stays $2$.\n\nNon-example: the mass $\\mathbf{M}$ is a norm on currents of finite mass, but it is not the flat norm, as the example shows.\n\n## Explanations\n\n### L1 human\n\nThe flat norm of a current is the cheapest way to write it as a piece plus a boundary, where each part costs its mass.\n",
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