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  "body": "\n## Summary\n\nGeometric measure theory studies the geometry of sets and measures in Euclidean space and in manifolds with the tools of measure theory.\nIts basic objects are [[Hausdorff measure|Q1TGZK8RN6]], [[rectifiable sets|7N0GFMW05R]], [[currents|360H7MD9RD]] and [[varifolds|N3QA80M44M]], which act as surfaces that may be singular, have multiplicities, or arise as limits.\nIt provides compactness theorems for such generalised surfaces, which give existence results for variational problems such as [[Plateau's problem|DVYA6GD8RN]] in every dimension and codimension.\nIt also studies the regularity of the minimisers and the structure of their singular sets.\n\n## Explanations\n\n### L1 human\n\nSoap films can have edges where three sheets meet, and limits of smooth surfaces can develop folds and corners.\nClassical calculus works with smooth surfaces and struggles with such objects.\nGeometric measure theory replaces a surface by something more flexible, such as a measure that records area, so that limits always exist.\nFor example, the graph of the function $\\abs{x}$ has a corner, but it still has a well-defined length and an approximate tangent line at almost every point.\nOnce a best generalised surface is found, one proves that it is smooth except on a small set.\n",
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