{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, let $V$ be a [[vector space|05EGHEQZAW]] over $\\mathbb{K}$, and let $K \\subseteq V$ be an [[absorbing set|NM1BCK3ZX9]].\nThe *Minkowski functional* (or *gauge*) of $K$ is the map $p_K \\colon V \\to [0, \\infty)$ given by\n$$p_K(x) = \\inf \\set{t \u003e 0 : x \\in t K}.$$\n\n## Remarks\n\nSince $K$ is absorbing, the set $\\set{t \u003e 0 : x \\in tK}$ is nonempty, so $p_K(x)$ is a finite number.\nIf $K$ is convex, then $p_K$ is sublinear: $p_K(x + y) \\le p_K(x) + p_K(y)$ and $p_K(s x) = s \\, p_K(x)$ for $s \\ge 0$.\nIf $K$ is convex and [[balanced|2TDSHJJYS4]], then $p_K$ is a [[seminorm|FP5GVCN8GV]].\nExample: let $\\norm{\\cdot}$ be a norm on $V$ and $K = \\set{x : \\norm{x} \\le 1}$; then $x \\in tK$ if and only if $\\norm{x} \\le t$, so $p_K(x) = \\norm{x}$.\nNon-example: in $\\R^2$ the segment $K = [-1, 1] \\times \\set{0}$ is not absorbing, and for $x = (0, 1)$ the set $\\set{t \u003e 0 : x \\in tK}$ is empty, so no finite value $p_K(x)$ is defined.\nSome authors allow any set $K$ and set $p_K(x) = \\infty$ when no $t$ exists.\nIn a topological vector space, the Minkowski functional of a convex open neighbourhood $K$ of $0$ is continuous, and $K = \\set{x : p_K(x) \u003c 1}$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "PAX9SHDHDW",
    "kind": "definition",
    "msc": [
      "46A03",
      "52A07"
    ],
    "name": "Minkowski functional",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Minkowski_functional"
      }
    ],
    "tag": "38CEJPFESJ",
    "uses": [
      "NM1BCK3ZX9",
      "05EGHEQZAW"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/38CEJPFESJ/",
  "links": [
    {
      "field": "defines",
      "target": "PAX9SHDHDW"
    },
    {
      "field": "uses",
      "target": "NM1BCK3ZX9"
    },
    {
      "field": "uses",
      "target": "05EGHEQZAW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/38CEJPFESJ.md",
  "msc": [
    "46A03",
    "52A07"
  ],
  "name": "Minkowski functional",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/38C/38CEJPFESJ.md",
  "status": null,
  "tag": "38CEJPFESJ",
  "uri": "m2://38CEJPFESJ",
  "url": "https://math2.tamnd.com/wiki/minkowski-functional-definition/",
  "writing": {
    "tag": "38CEJPFESJ",
    "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
      }
    ]
  }
}