{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "7V2X68AMEY"
    },
    {
      "field": "uses",
      "target": "7V2X68AMEY"
    },
    {
      "field": "defines",
      "target": "PM71JAZV31"
    }
  ],
  "body": "\n## Summary\n\nA valuation on convex bodies is a function $\\varphi$ on the nonempty compact convex sets of $\\R^n$ that is additive in the inclusion–exclusion sense.\nThis means $\\varphi(K \\cup L) + \\varphi(K \\cap L) = \\varphi(K) + \\varphi(L)$ whenever $K \\cup L$ is convex.\nVolume, surface area, the Euler characteristic and all [[intrinsic volumes|C2CK6KSSTV]] are valuations.\nValuations are the finitely additive \"measures\" of convex geometry, and their classification, starting with [[Hadwiger's theorem|7V2X68AMEY]], is a central theme of integral geometry.\nThis notion is unrelated to valuations of fields in algebra.\n\n## Explanations\n\n### L1 human\n\nA valuation is a way to measure convex shapes that does not count overlaps twice.\nIf two convex shapes together form a convex shape, then the measure of the union plus the measure of the overlap equals the sum of the two measures.\nArea and perimeter of plane convex sets behave like this.\nThe function that gives $1$ to every nonempty convex set also does: $1 + 1 = 1 + 1$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "52B45",
      "52A20"
    ],
    "name": "Valuation on convex bodies",
    "same_as": {
      "wikidata": "Q85812715",
      "wikipedia": "Valuation (geometry)"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Valuation_(geometry)"
      }
    ],
    "tag": "V0RBKHYTV7"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/V0RBKHYTV7/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/V0RBKHYTV7.md",
  "msc": [
    "52B45",
    "52A20"
  ],
  "name": "Valuation on convex bodies",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/V0R/V0RBKHYTV7.md",
  "status": null,
  "tag": "V0RBKHYTV7",
  "uri": "m2://V0RBKHYTV7",
  "url": "https://tamnd.github.io/math2/wiki/valuation-on-convex-bodies/",
  "writing": {
    "tag": "V0RBKHYTV7",
    "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
      }
    ]
  }
}