{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "H5DEZ7Q1A6"
    },
    {
      "field": "defines",
      "target": "HN5A4M2GE8"
    }
  ],
  "body": "\n## Summary\n\nThe Minkowski sum of two convex polytopes $P$ and $Q$ is the set of all sums $p + q$ with $p \\in P$ and $q \\in Q$.\nIt is again a [[convex polytope|300K6CB1KH]], and it is the special case of [[Minkowski addition|SXS6D87PHE]] for polytopes.\nIts faces are sums of faces of $P$ and $Q$.\nIts [[normal fan|XJ83PTS3H0]] is the common refinement of the normal fans of $P$ and $Q$.\nMinkowski sums lead to [[zonotopes|Z5SA6A67KF]], to [[mixed volumes|QB3Q5FBJ08]] and to the Newton polytopes of products of polynomials.\n\n## Explanations\n\n### L1 human\n\nSlide a copy of $Q$ so that a chosen point of it visits every point of $P$, and collect everything that $Q$ sweeps out.\nThe swept region is the Minkowski sum $P + Q$.\nFor example, the sum of a horizontal unit segment and a vertical unit segment is the unit square.\nThe sum of a triangle and the same triangle turned upside down is a hexagon.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "52B11",
      "52A39"
    ],
    "name": "Minkowski sum of polytopes",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Minkowski_addition"
      }
    ],
    "specializes": [
      "SXS6D87PHE"
    ],
    "tag": "7GK5417BJF"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/7GK5417BJF/",
  "links": [
    {
      "field": "specializes",
      "target": "SXS6D87PHE"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/7GK5417BJF.md",
  "msc": [
    "52B11",
    "52A39"
  ],
  "name": "Minkowski sum of polytopes",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7GK/7GK5417BJF.md",
  "status": null,
  "tag": "7GK5417BJF",
  "uri": "m2://7GK5417BJF",
  "url": "https://tamnd.github.io/math2/wiki/minkowski-sum-of-polytopes/",
  "writing": {
    "tag": "7GK5417BJF",
    "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
      }
    ]
  }
}