{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a real [[normed space|03J5RAKB30]] with continuous [[dual space|07JBSZE0V4]] $X^*$, and write $\\ip{y}{x} = y(x)$ for $y \\in X^*$ and $x \\in X$.\nLet $f \\colon X \\to [-\\infty, +\\infty]$ be a function.\nThe *Fenchel conjugate* (or *convex conjugate*) of $f$ is the function $f^* \\colon X^* \\to [-\\infty, +\\infty]$ given by\n$$f^*(y) = \\sup_{x \\in X} \\big(\\ip{y}{x} - f(x)\\big),$$\nwhere $\\sup$ is the [[supremum|0R3H5PANRQ]] in the [[extended real line|0GSZ7B0CGR]].\nFor $X = \\R^n$, one identifies $X^*$ with $\\R^n$ through the dot product.\n\n## Remarks\n\nExample: $f(x) = \\frac12 x^2$ on $\\R$.\nFor fixed $y$, the function $x \\mapsto yx - \\frac12 x^2$ is maximal at $x = y$, so $f^*(y) = \\frac12 y^2$.\n\nExample without differentiability: $f(x) = \\abs{x}$ on $\\R$.\nIf $\\abs{y} \\le 1$ then $yx - \\abs{x} \\le 0$ with equality at $x = 0$, so $f^*(y) = 0$; if $\\abs{y} \u003e 1$ then $yx - \\abs{x} \\to +\\infty$ along $x = t \\sgn(y)$, so $f^*(y) = +\\infty$.\nThe classical formula of the [[Legendre transformation|ZQYMWVT1TT]] does not apply to this $f$, because $f$ is not differentiable at $0$.\n\nNon-example of the identity $f^{**} = f$: for $f(x) = -x^2$ on $\\R$, every $f^*(y)$ is $+\\infty$, so $f^{**} \\equiv -\\infty \\neq f$; this $f$ is not convex.\n\nEvery conjugate $f^*$ is convex and lower semicontinuous, as a supremum of affine continuous functions.\nThe Fenchel–Young inequality $f(x) + f^*(y) \\ge \\ip{y}{x}$ holds whenever the left side is defined.\nThe Fenchel–Moreau theorem states that $f^{**} = f$ on $X$ when $f$ is proper, convex and lower semicontinuous.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "T7V2VPY61S",
    "kind": "definition",
    "msc": [
      "52A41",
      "26B25",
      "90C46"
    ],
    "name": "Fenchel conjugate",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Convex_conjugate"
      }
    ],
    "tag": "B1X92F7SJ6",
    "uses": [
      "03J5RAKB30",
      "07JBSZE0V4",
      "0R3H5PANRQ",
      "0GSZ7B0CGR"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/B1X92F7SJ6/",
  "links": [
    {
      "field": "defines",
      "target": "T7V2VPY61S"
    },
    {
      "field": "uses",
      "target": "03J5RAKB30"
    },
    {
      "field": "uses",
      "target": "07JBSZE0V4"
    },
    {
      "field": "uses",
      "target": "0R3H5PANRQ"
    },
    {
      "field": "uses",
      "target": "0GSZ7B0CGR"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/B1X92F7SJ6.md",
  "msc": [
    "52A41",
    "26B25",
    "90C46"
  ],
  "name": "Fenchel conjugate",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/B1X/B1X92F7SJ6.md",
  "status": null,
  "tag": "B1X92F7SJ6",
  "uri": "m2://B1X92F7SJ6",
  "url": "https://tamnd.github.io/math2/wiki/fenchel-conjugate-definition/",
  "writing": {
    "tag": "B1X92F7SJ6",
    "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
      }
    ]
  }
}