{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 2$, let $K \\subseteq \\R^n$ be a [[convex body|437FSG3419]], and let $u \\in \\R^n$ be a unit vector.\nLet $u^\\perp = \\set{x \\in \\R^n : x \\cdot u = 0}$.\nFor $x \\in u^\\perp$, the set $K \\cap \\set{x + t u : t \\in \\R}$ is empty or a closed segment; let $\\ell(x)$ be its length when it is nonempty.\nThe *Steiner symmetrization* of $K$ in the direction $u$ is the set\n$$S_u K = \\set{x + t u : x \\in u^\\perp, \\ K \\cap (x + \\R u) \\ne \\emptyset, \\ \\abs{t} \\le \\tfrac{1}{2} \\ell(x)}.$$\n\n## Remarks\n\nThe set $S_u K$ is again a convex body, it is symmetric under the reflection in $u^\\perp$, and it has the same volume as $K$.\nIts surface area and its diameter are at most those of $K$.\nExample: let $K$ be the triangle with vertices $(0, 0)$, $(2, 0)$, $(0, 2)$ in $\\R^2$ and $u = (0, 1)$.\nThe vertical chord over $x \\in [0, 2]$ has length $2 - x$.\nSo $S_u K = \\set{(x, t) : 0 \\le x \\le 2, \\ \\abs{t} \\le (2 - x)/2}$, the triangle with vertices $(0, 1)$, $(0, -1)$, $(2, 0)$, of area $2$ like $K$.\nNon-example: the set $\\set{x + t u : 0 \\le t \\le \\ell(x)}$ has the same chord lengths as $S_u K$, but its chords are not centred on $u^\\perp$, so it is not the Steiner symmetrization.\nSome authors write $\\mathrm{st}_u K$ for $S_u K$; the operation is also defined for compact sets, with $\\ell(x)$ the one-dimensional measure of the section.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "8BMM456H7G",
    "kind": "definition",
    "msc": [
      "52A20",
      "52A40"
    ],
    "name": "Steiner symmetrization",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Isoperimetric_inequality"
      }
    ],
    "tag": "3YF7P33WRM",
    "uses": [
      "437FSG3419"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/3YF7P33WRM/",
  "links": [
    {
      "field": "defines",
      "target": "8BMM456H7G"
    },
    {
      "field": "uses",
      "target": "437FSG3419"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/3YF7P33WRM.md",
  "msc": [
    "52A20",
    "52A40"
  ],
  "name": "Steiner symmetrization",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3YF/3YF7P33WRM.md",
  "status": null,
  "tag": "3YF7P33WRM",
  "uri": "m2://3YF7P33WRM",
  "url": "https://tamnd.github.io/math2/wiki/steiner-symmetrization-definition/",
  "writing": {
    "tag": "3YF7P33WRM",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}