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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $E \\subseteq \\R^n$ be a set of locally finite [[perimeter|Y68BHXZJBV]], that is, a local [[Caccioppoli set|W6R7QNEAF1]].\nLet $\\mu = \\abs{D\\chi_E}$ be the total variation measure of the distributional gradient of its indicator function $\\chi_E$, a [[Radon measure|HZNEQQBCW4]].\nWrite $D\\chi_E = -\\nu_E \\, \\mu$ with a Borel map $\\nu_E \\colon \\R^n \\to \\R^n$ such that $\\abs{\\nu_E} = 1$ $\\mu$-almost everywhere.\nFor $x \\in \\R^n$ and $r \u003e 0$ let $B_r(x)$ be the open ball of radius $r$ about $x$.\nConsider the limit\n$$\\lim_{r \\to 0} \\frac{D\\chi_E(B_r(x))}{\\mu(B_r(x))}.$$\nThe *reduced boundary* $\\partial^* E$ is the set of points $x$ in the support of $\\mu$ at which this limit exists and has norm $1$.\nLet $\\mathcal{H}^{n-1}$ be the $(n-1)$-dimensional [[Hausdorff measure|Q1TGZK8RN6]].\nThen:\n\n1. The set $\\partial^* E$ is countably $(n-1)$-[[rectifiable|7N0GFMW05R]]: up to an $\\mathcal{H}^{n-1}$-null set, it is contained in a countable union of $C^1$ hypersurfaces.\n2. $\\mu = \\mathcal{H}^{n-1} \\llcorner \\partial^* E$, that is, $\\mu(A) = \\mathcal{H}^{n-1}(A \\cap \\partial^* E)$ for every Borel set $A$.\n3. For every $x \\in \\partial^* E$, the rescaled sets $(E - x)/r$ converge in $L^1_{\\mathrm{loc}}(\\R^n)$, as $r \\to 0$, to the half-space $\\set{y : y \\cdot \\nu_E(x) \u003c 0}$.\n4. For every $\\varphi \\in C^1_c(\\R^n; \\R^n)$, the generalized Gauss–Green formula holds:\n$$\\int_E \\operatorname{div} \\varphi \\, dx = \\int_{\\partial^* E} \\varphi \\cdot \\nu_E \\, d\\mathcal{H}^{n-1}.$$\n\n## Remarks\n\nThe proof starts from the Besicovitch differentiation theorem, which shows that $\\mu$-almost every point lies in $\\partial^* E$.\nAt a point of $\\partial^* E$, the isoperimetric inequality gives density estimates comparing $\\mu(B_r(x))$ with $r^{n-1}$.\nWith the compactness of sets of finite perimeter, a blow-up argument then shows that the rescaled sets converge to a half-space.\nThe half-space limit identifies $\\mu$ with $\\mathcal{H}^{n-1}$ on $\\partial^* E$.\nRectifiability then follows from the existence of approximate tangent planes at all points of $\\partial^* E$.\nFor a bounded open set with $C^1$ boundary, $\\partial^* E = \\partial E$, $\\nu_E$ is the outer unit normal, and item 4 is the classical divergence theorem.\nFor the square $[0, 1]^2$, the reduced boundary is the boundary minus the four corners; the corners form an $\\mathcal{H}^1$-null set.\nThe topological boundary can be much larger than $\\partial^* E$, even of positive Lebesgue measure, so the reduced boundary is the right notion of boundary here.\n\n## Explanations\n\n### L1 human\n\nA set of finite perimeter can have a very rough boundary.\nThe theorem says that the part of the boundary that matters is still made of pieces of smooth surfaces, with a well-defined outward direction at almost every point.\nZooming in at such a point, the set looks more and more like half of space.\nWith this boundary, the usual formula relating the flux through the boundary to the integral of the divergence still holds.\n\n### L2 human\n\nThe theorem is due to Ennio De Giorgi, and it is the foundation of the theory of sets of finite perimeter.\nIt turns a set known only through the measure $D\\chi_E$ into a geometric object with a measure-theoretic normal.\nFederer later showed that $\\partial^* E$ coincides, up to an $\\mathcal{H}^{n-1}$-null set, with the essential boundary, the set of points where $E$ has density neither $0$ nor $1$.\nThese results are the starting point of the regularity theory of perimeter minimizers and of the study of free discontinuity problems.\n",
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