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  "body": "\n## Summary\n\nSteiner symmetrization is an operation that turns a [[convex body|437FSG3419]] into a convex body that is symmetric about a chosen hyperplane.\nIt slides each chord of the body orthogonal to the hyperplane along its line until the hyperplane cuts the chord in its midpoint.\nThe operation keeps the volume, and it does not increase the surface area or the diameter.\nRepeating it in well chosen directions makes a body converge to a ball, which gives proofs of the [[isoperimetric inequality|KBX8RBZ626]] and of related extremal results.\n\n## Explanations\n\n### L1 human\n\nThink of a convex body as a bundle of straight sticks, all parallel to one fixed direction.\nSlide every stick along its own line until its midpoint lies on one fixed plane.\nThe sticks keep their lengths, so the volume does not change.\nThe new body is a mirror image of itself across the plane, and it is \"rounder\" than before.\nFor example, the right triangle with vertices $(0, 0)$, $(2, 0)$, $(0, 2)$, symmetrized about the $x$-axis, becomes the triangle with vertices $(0, 1)$, $(0, -1)$, $(2, 0)$, which has the same area $2$.\n",
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