{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $P$ be a nonempty [[convex polytope|300K6CB1KH]] in $\\R^n$, and write $\\ip{u}{x}$ for the standard inner product.\nFor a nonempty [[face|KY7SB854VH]] $F$ of $P$, the *normal cone* of $F$ is\n$$N_F = \\set{u \\in \\R^n : \\ip{u}{x} \\ge \\ip{u}{y} \\text{ for all } x \\in F \\text{ and } y \\in P}.$$\nSo $N_F$ is the set of vectors $u$ such that the linear functional $\\ip{u}{\\cdot}$ attains its maximum over $P$ at every point of $F$.\nThe *normal fan* of $P$ is the collection $\\mathcal{N}(P) = \\set{N_F : F \\text{ a nonempty face of } P}$.\n\n## Remarks\n\nEach $N_F$ is a polyhedral [[convex cone|6VE4KA12VD]], and $\\dim N_F = n - \\dim F$.\nThe collection $\\mathcal{N}(P)$ is a complete fan:\n\n1. every face of a cone of $\\mathcal{N}(P)$ is a cone of $\\mathcal{N}(P)$;\n2. the intersection of two cones of $\\mathcal{N}(P)$ is a face of each of them;\n3. the union of all cones of $\\mathcal{N}(P)$ is $\\R^n$.\n\nInclusion is reversed: $F \\subseteq G$ holds exactly when $N_G \\subseteq N_F$.\nEach $N_F$ contains the orthogonal complement of the direction space of the affine hull of $P$.\nSo the cones are pointed exactly when $P$ is full-dimensional, and then the vertices of $P$ give the $n$-dimensional cones and the facets give the rays.\n\nExample: let $P = [-1,1]^2$.\nThe vertex $(1,1)$ maximizes $\\ip{u}{\\cdot}$ exactly when $u_1 \\ge 0$ and $u_2 \\ge 0$, so its normal cone is the closed first quadrant.\nThe edge $\\set{1} \\times [-1,1]$ is maximal exactly for $u_1 \\ge 0$ and $u_2 = 0$, so its normal cone is the ray spanned by $(1,0)$.\nThe face $P$ itself has normal cone $\\set{0}$.\nSo $\\mathcal{N}(P)$ consists of the four closed quadrants, the four half-axes and the origin.\n\nNon-example: the closed upper half-plane and the closed right half-plane do not form a fan, even with all their faces added.\nTheir intersection is the first quadrant, which is not a face of either half-plane, so clause 2 fails.\n\nSome authors use the functionals that are minimal on $F$ instead; this replaces each cone by its negative.\nThe normal fan of a Minkowski sum $P + Q$ is the common refinement of the normal fans of $P$ and $Q$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "XJ83PTS3H0",
    "kind": "definition",
    "msc": [
      "52B11",
      "14M25"
    ],
    "name": "Normal fan",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Toric_variety"
      }
    ],
    "tag": "7H0E1ZBWTB",
    "uses": [
      "300K6CB1KH",
      "KY7SB854VH",
      "6VE4KA12VD"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/7H0E1ZBWTB/",
  "links": [
    {
      "field": "defines",
      "target": "XJ83PTS3H0"
    },
    {
      "field": "uses",
      "target": "300K6CB1KH"
    },
    {
      "field": "uses",
      "target": "KY7SB854VH"
    },
    {
      "field": "uses",
      "target": "6VE4KA12VD"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/7H0E1ZBWTB.md",
  "msc": [
    "52B11",
    "14M25"
  ],
  "name": "Normal fan",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7H0/7H0E1ZBWTB.md",
  "status": null,
  "tag": "7H0E1ZBWTB",
  "uri": "m2://7H0E1ZBWTB",
  "url": "https://tamnd.github.io/math2/wiki/normal-fan-definition/",
  "writing": {
    "tag": "7H0E1ZBWTB",
    "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
      }
    ]
  }
}