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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $K \\subseteq \\R^n$ be a compact [[convex body|437FSG3419]] with nonempty interior.\nAn ellipsoid is a set $c + AB$, with $B$ the closed unit ball, $c \\in \\R^n$ and $A$ an invertible real $n \\times n$ [[matrix|0VD9ZSWWJG]]; volume means [[Lebesgue measure|2Z2QVWZ262]].\n\n1. Among the ellipsoids contained in $K$ there is exactly one, $E = c + AB$, of maximal volume.\n2. For this $E$, one has $K \\subseteq c + nAB$.\n3. If moreover $K = -K$, then $c = 0$ and $K \\subseteq \\sqrt{n}\\, E$.\n4. Among the ellipsoids containing $K$ there is exactly one of minimal volume.\n\n## Remarks\n\nOne may take $A$ symmetric positive definite, by the polar decomposition.\nExistence follows from compactness: the pairs $(c, A)$ with $A$ positive semidefinite and $c + AB \\subseteq K$ form a compact set, and volume is continuous.\nFor uniqueness, let $c_1 + A_1 B$ and $c_2 + A_2 B$, with $A_i$ positive definite, both have maximal volume.\nThen $\\frac{c_1 + c_2}{2} + \\frac{A_1 + A_2}{2} B$ also lies in $K$.\nIts volume is strictly larger unless $A_1 = A_2$, by the strict log-concavity of the determinant on positive definite matrices.\nIf $A_1 = A_2$ but $c_1 \\neq c_2$, the convex hull of the two ellipsoids contains a larger ellipsoid, stretched in the direction $c_2 - c_1$.\nJohn characterized the case $E = B$ by contact points.\nFor $K \\supseteq B$, the ball $B$ is the maximal ellipsoid if and only if there are $u_1, \\dots, u_m \\in \\partial K$ with $\\abs{u_i} = 1$ and weights $c_i \u003e 0$ such that $\\sum_i c_i u_i = 0$ and $\\sum_i c_i u_i u_i^T = I_n$.\nThe bounds in 2 and 3 follow from this.\nThe factor $n$ in 2 is attained by a regular simplex.\nThe factor $\\sqrt{n}$ in 3 is attained by the cube $[-1, 1]^n$, whose maximal ellipsoid is $B$ and which contains the point $(1, \\dots, 1)$ of norm $\\sqrt{n}$.\n\n## Explanations\n\n### L1 human\n\nEvery convex shape with some thickness contains a largest ellipsoid, and it is unique.\nThe shape is then caught between this ellipsoid and the same ellipsoid enlarged by a factor equal to the dimension.\nFor shapes that are symmetric about a center, the factor improves to the square root of the dimension.\nSo, after a linear change of coordinates, every convex body looks like a ball up to a bounded distortion.\n\n### L2 human\n\nPart 3 says that every $n$-dimensional normed space is at [[Banach–Mazur distance|ZJKSYYASPT]] at most $\\sqrt{n}$ from Euclidean space, a basic fact of the local theory of Banach spaces.\nThe contact point condition in John's characterization is a decomposition of the identity.\nBall combined it with the Brascamp–Lieb inequality to prove his reverse isoperimetric inequality.\nIn optimization, rounding algorithms for convex bodies compute approximate John ellipsoids.\n",
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