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  "body": "\n## Summary\n\nThe John ellipsoid of a convex body $K$ in $\\R^n$ with nonempty interior is the ellipsoid of largest volume contained in $K$.\nIt exists and is unique by the [[Löwner–John theorem|TM5SYH4XHY]], and $K$ is contained in its dilate by the factor $n$ about its center.\nIt gives a canonical affine position of $K$.\nIt is the starting point of many estimates in convex geometry and the geometry of normed spaces.\nIts dual partner is the Löwner ellipsoid, the ellipsoid of least volume containing $K$.\n\n## Explanations\n\n### L1 human\n\nInside any convex shape one can fit ellipses (or ellipsoids, in space) of many sizes and directions.\nAmong them there is exactly one of largest area: the John ellipsoid.\nFor a square it is the inscribed circle, and for a long rectangle it is the inscribed ellipse that touches all four sides at their midpoints.\nEvery convex shape is squeezed between its John ellipsoid and a copy of it blown up by a factor equal to the dimension.\n",
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