{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 1$.\nA *zonoid* in $\\R^n$ is a [[convex body|437FSG3419]] $Z$ that is the limit of a sequence of [[zonotopes|Z5SA6A67KF]] in $\\R^n$.\nThe limit is taken in the [[Hausdorff distance|F79WYYC07C]].\nHere a zonotope is a Minkowski sum $[a_1, b_1] + \\cdots + [a_m, b_m]$ of finitely many closed segments.\n\n## Remarks\n\nEquivalently, the [[support function|1H92CYA4ZC]] of $Z$ has the form $h_Z(x) = x \\cdot a + \\int_{S^{n-1}} \\abs{x \\cdot v} \\, d\\mu(v)$.\nHere $a \\in \\R^n$, and $\\mu$ is a finite even Borel measure on the unit sphere $S^{n-1}$.\nExample: the closed unit ball $B$ of $\\R^n$ is a zonoid.\nIts support function is $h_B(x) = \\abs{x}$, and by rotation invariance $\\int_{S^{n-1}} \\abs{x \\cdot v} \\, d\\sigma(v) = c_n \\abs{x}$ for the normalized surface measure $\\sigma$ and a constant $c_n \u003e 0$, so $\\mu = \\sigma / c_n$ works with $a = 0$.\nNon-example: a triangle in $\\R^2$ is not a zonoid.\nEach zonotope is symmetric about a point, this property passes to Hausdorff limits, and a triangle has no centre of symmetry.\nIn $\\R^2$ every convex body with a centre of symmetry is a zonoid; in $\\R^3$ the regular octahedron is a centrally symmetric convex body that is not a zonoid.\nSome authors also call lower-dimensional Hausdorff limits of zonotopes, such as single segments, zonoids; here a zonoid is required to have nonempty interior, like every convex body.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "ZMZZ2QRFTZ",
    "kind": "definition",
    "msc": [
      "52A21",
      "52A20"
    ],
    "name": "Zonoid",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Zonoid"
      }
    ],
    "tag": "7Z3VD04JGX",
    "uses": [
      "437FSG3419",
      "Z5SA6A67KF",
      "F79WYYC07C",
      "1H92CYA4ZC"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/7Z3VD04JGX/",
  "links": [
    {
      "field": "defines",
      "target": "ZMZZ2QRFTZ"
    },
    {
      "field": "uses",
      "target": "437FSG3419"
    },
    {
      "field": "uses",
      "target": "Z5SA6A67KF"
    },
    {
      "field": "uses",
      "target": "F79WYYC07C"
    },
    {
      "field": "uses",
      "target": "1H92CYA4ZC"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/7Z3VD04JGX.md",
  "msc": [
    "52A21",
    "52A20"
  ],
  "name": "Zonoid",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7Z3/7Z3VD04JGX.md",
  "status": null,
  "tag": "7Z3VD04JGX",
  "uri": "m2://7Z3VD04JGX",
  "url": "https://tamnd.github.io/math2/wiki/zonoid-definition/",
  "writing": {
    "tag": "7Z3VD04JGX",
    "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
      }
    ]
  }
}