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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $B$ be the closed unit ball of $\\R^n$.\nAn *ellipsoid* in $\\R^n$ is a set $E = c + A B = \\set{c + Ax : \\abs{x} \\le 1}$, where $c \\in \\R^n$ and $A$ is an invertible real $n \\times n$ [[matrix|0VD9ZSWWJG]].\nIts volume is $\\abs{\\det A}\\, \\kappa_n$, where $\\kappa_n$ is the [[Lebesgue measure|2Z2QVWZ262]] of $B$.\nLet $K \\subseteq \\R^n$ be a compact [[convex body|437FSG3419]] with nonempty interior.\nThe *John ellipsoid* of $K$ is the ellipsoid of maximal volume among all ellipsoids $E$ with $E \\subseteq K$.\n\n## Remarks\n\nThe [[Löwner–John theorem|TM5SYH4XHY]] shows that this ellipsoid exists and is unique.\nThe ellipsoid of minimal volume containing $K$ is the *Löwner ellipsoid* (or Löwner–John ellipsoid) of $K$.\n\nExample: $K = [-1, 1]^2$.\nIts John ellipsoid is the unit disk $B$.\nThe problem is invariant under the symmetries of the square, so by uniqueness the maximal ellipse is invariant under them too.\nAn ellipse invariant under rotation by $90°$ about the origin is a disk centered at $0$, and the largest such disk in $K$ is $B$.\n\nNon-example: in the rectangle $K = [-2, 2] \\times [-1, 1]$ the unit disk lies in $K$ but is not the John ellipsoid.\nThe ellipse $\\set{(x, y) : x^2/4 + y^2 \\le 1}$ also lies in $K$ and has area $2\\pi \u003e \\pi$.\n\nThe condition that $K$ has nonempty interior is needed: otherwise every ellipsoid in $K$ would be degenerate, and no ellipsoid in the above sense lies in $K$.\n",
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