{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "7Z3VD04JGX"
    },
    {
      "field": "uses",
      "target": "PE2HN8MY2J"
    }
  ],
  "body": "\n## Summary\n\nA zonoid is a [[convex body|437FSG3419]] that is a limit, in the [[Hausdorff distance|F79WYYC07C]], of [[zonotopes|Z5SA6A67KF]], that is, of finite Minkowski sums of line segments.\nZonoids are exactly the convex bodies whose support functions are integrals of the functions $x \\mapsto \\abs{x \\cdot v}$, up to a translation.\nThey appear in the geometry of finite-dimensional normed spaces and in the study of projections of convex bodies.\nBy Lyapunov's convexity theorem, they also arise as the ranges of nonatomic vector-valued measures.\n\n## Explanations\n\n### L1 human\n\nAdding line segments in the sense of Minkowski gives polytopes with many parallel edges, such as parallelograms, hexagons with parallel opposite sides, or cubes.\nA zonoid is what one gets by adding more and more, ever shorter segments, and passing to the limit.\nThe round disc is a zonoid: it is the limit of regular polygons with $2m$ sides, each of which is a sum of $m$ segments.\nEvery zonoid is centrally symmetric, so a triangle is never a zonoid.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "52A21",
      "52A20"
    ],
    "name": "Zonoid",
    "same_as": {
      "wikidata": "Q131417845",
      "wikipedia": "Zonoid"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Zonoid"
      }
    ],
    "specializes": [
      "437FSG3419"
    ],
    "tag": "ZMZZ2QRFTZ"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/ZMZZ2QRFTZ/",
  "links": [
    {
      "field": "specializes",
      "target": "437FSG3419"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/ZMZZ2QRFTZ.md",
  "msc": [
    "52A21",
    "52A20"
  ],
  "name": "Zonoid",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZMZ/ZMZZ2QRFTZ.md",
  "status": null,
  "tag": "ZMZZ2QRFTZ",
  "uri": "m2://ZMZZ2QRFTZ",
  "url": "https://tamnd.github.io/math2/wiki/zonoid/",
  "writing": {
    "tag": "ZMZZ2QRFTZ",
    "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
      }
    ]
  }
}