{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "B7X9M15Q92"
    },
    {
      "field": "defines",
      "target": "J3MJEDDHNY"
    }
  ],
  "body": "\n## Summary\n\nThe decreasing rearrangement of a measurable function $f$ is the non-increasing, right-continuous function $f^*$ on $[0, \\infty)$ that has the same distribution function as $\\abs{f}$.\nIt sorts the values of $\\abs{f}$ from the largest to the smallest.\nEvery norm that depends only on the distribution function, such as the $L^p$ norms, takes the same value on $f$ and on $f^*$.\nIt is the basic tool of Lorentz spaces and of rearrangement-invariant function spaces.\n\n## Explanations\n\n### L1 human\n\nImagine the values of a function as piles of sand placed along a line.\nThe decreasing rearrangement moves the piles so that the tallest come first and the heights go down from left to right.\nNo sand is added or removed, so the amount of land above each height stays the same.\nFor example, on $[0, 1]$ the function $f(x) = x$ has decreasing rearrangement $f^*(t) = 1 - t$ for $0 \\le t \u003c 1$ and $f^*(t) = 0$ for $t \\ge 1$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "46E30",
      "28A20"
    ],
    "name": "Decreasing rearrangement",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Symmetric_decreasing_rearrangement"
      }
    ],
    "tag": "BWS042NC2P"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/BWS042NC2P/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/BWS042NC2P.md",
  "msc": [
    "46E30",
    "28A20"
  ],
  "name": "Decreasing rearrangement",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BWS/BWS042NC2P.md",
  "status": null,
  "tag": "BWS042NC2P",
  "uri": "m2://BWS042NC2P",
  "url": "https://math2.tamnd.com/wiki/decreasing-rearrangement/",
  "writing": {
    "tag": "BWS042NC2P",
    "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
      }
    ]
  }
}