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  "body": "\n## Statement\n\nLet $k \\ge 1$ be an integer and let $\\varepsilon \u003e 0$.\nThen there is an integer $N = N(k, \\varepsilon)$ with the following property.\nLet $X$ be a real [[normed space|03J5RAKB30]] of dimension at least $N$.\nThen $X$ has a $k$-dimensional [[linear subspace|00YC1D652K]] $E$ with a linear map $T \\colon \\R^k \\to E$ such that\n$$\\abs{x} \\le \\norm{T x} \\le (1 + \\varepsilon) \\abs{x} \\quad \\text{for all } x \\in \\R^k,$$\nwhere $\\abs{x}$ is the Euclidean norm of $x$.\n\n## Remarks\n\nThe theorem was proved by Dvoretzky.\nV. Milman gave a proof based on concentration of measure on the high-dimensional sphere.\nHis proof shows that one can take $k$ of order $c(\\varepsilon) \\log n$ in a space of dimension $n$.\nThe idea: a norm, seen as a Lipschitz function on the Euclidean sphere, is almost constant on most of the sphere.\nA random $k$-dimensional subspace then sees only this almost constant part.\n\nThe bound $\\log n$ cannot be improved in general.\nFor $\\ell_\\infty^n$, that is $\\R^n$ with the max norm, the largest $k$ with a subspace at distance at most $2$ from Euclidean is of order $\\log n$.\nFor $\\ell_1^n$, in contrast, there are almost Euclidean subspaces of dimension proportional to $n$.\n\n## Explanations\n\n### L1 human\n\nEvery convex body that is symmetric about the origin and lives in very many dimensions has a slice through the center that is almost a round ball.\nThe slice can be large if the dimension is large.\nSo round balls, that is Euclidean geometry, appear inside every normed space of high dimension.\n\n### L2 human\n\nIn the language of the local theory of Banach spaces, the theorem says that $\\ell_2$ is finitely representable in every infinite-dimensional Banach space.\nIt is a starting point of asymptotic geometric analysis, together with the notions of [[type and cotype|BF9XR5SVPV]].\nThe distance used here is the Banach–Mazur distance of the [[Banach–Mazur compactum|RH1WTN38RQ]].\n",
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