{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "E7TY2TPXX7"
    },
    {
      "field": "defines",
      "target": "YJ2VVJZBX4"
    }
  ],
  "body": "\n## Summary\n\nThe symmetric decreasing rearrangement of a function $f$ on $\\R^n$ is a radial function $f^*$, decreasing in $\\abs{x}$.\nIts superlevel sets are balls centred at the origin with the same volumes as the superlevel sets of $\\abs{f}$.\nIt keeps every $L^p$ norm, does not increase the Dirichlet energy, and does not decrease certain double integrals.\nSo it is the main tool to show that balls and radial functions are optimal in variational problems.\nExamples are the [[Faber–Krahn inequality|E7TY2TPXX7]] and the isoperimetric inequality.\nIt is the $n$-dimensional, symmetric version of the [[decreasing rearrangement|BWS042NC2P]] on $[0, \\infty)$.\n\n## Explanations\n\n### L1 human\n\nThink of the graph of $\\abs{f}$ as a mountain landscape made of horizontal slices.\nReplace each slice by a disc of the same area, centred over the origin, and stack the discs.\nThe result is a single round mountain with its peak at the origin and the same amount of material at each height.\nFor example, the indicator function of the interval $[1, 3]$ becomes the indicator function of $(-1, 1)$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "26D15",
      "46E30",
      "35B06"
    ],
    "name": "Symmetric decreasing rearrangement",
    "same_as": {
      "wikidata": "Q2502733",
      "wikipedia": "Symmetric decreasing rearrangement"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Symmetric_decreasing_rearrangement"
      }
    ],
    "tag": "2GRPCP5XK9"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/2GRPCP5XK9/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/2GRPCP5XK9.md",
  "msc": [
    "26D15",
    "46E30",
    "35B06"
  ],
  "name": "Symmetric decreasing rearrangement",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2GR/2GRPCP5XK9.md",
  "status": null,
  "tag": "2GRPCP5XK9",
  "uri": "m2://2GRPCP5XK9",
  "url": "https://tamnd.github.io/math2/wiki/symmetric-decreasing-rearrangement/",
  "writing": {
    "tag": "2GRPCP5XK9",
    "kind": "concept",
    "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
      }
    ]
  }
}