{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $V$ be a real [[vector space|000AYPA9AJ]] and let $C \\subseteq V$ be a [[convex set|02AS45T3H9]].\nA *face* of $C$ is a subset $F \\subseteq C$ such that:\n\n1. $F$ is convex;\n2. $F$ is an [[extreme set|WD3RJZC5A7]] of $C$: if $x_1, x_2 \\in C$ and the [[open segment|CCZA91NE5X]] $(x_1, x_2)$ meets $F$, then $x_1, x_2 \\in F$.\n\n## Remarks\n\nThe empty set and $C$ are faces of $C$.\nA one-point set $\\set{x}$ is a face exactly when $x$ is an [[extreme point|3MSGFJM2NX]] of $C$.\nA face of a face of $C$ is a face of $C$.\n\nExample: for the square $C = [0, 1]^2$, the edge $F = [0, 1] \\times \\set{1}$ is a face.\nIt is convex.\nIf a point of $F$ lies strictly between $x_1, x_2 \\in C$, the second coordinates of $x_1, x_2$ are at most $1$ and average to $1$ with positive weights.\nSo both equal $1$, and $x_1, x_2 \\in F$.\n\nNon-example: the lower half $F = [0, 1] \\times [0, 1/2]$ of the same square is convex, but the point $(1/2, 1/2) \\in F$ lies strictly between $(1/2, 0)$ and $(1/2, 1)$, and $(1/2, 1) \\notin F$, so clause 2 fails.\n\nIf $H$ is a [[supporting hyperplane|K7N565NKCV]] of $C$, then $C \\cap H$ is a face, called an exposed face.\nNot every face is exposed: in the stadium, the convex hull of two unit discs centred at $(\\pm 1, 0)$, the point $(1, 1)$ is a face but not an exposed face.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "KY7SB854VH",
    "kind": "definition",
    "msc": [
      "52A20",
      "52A05"
    ],
    "name": "Face of a convex set",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Face_(geometry)"
      }
    ],
    "tag": "7VPJQNQV34",
    "uses": [
      "000AYPA9AJ",
      "02AS45T3H9",
      "WD3RJZC5A7",
      "CCZA91NE5X",
      "3MSGFJM2NX",
      "K7N565NKCV"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/7VPJQNQV34/",
  "links": [
    {
      "field": "defines",
      "target": "KY7SB854VH"
    },
    {
      "field": "uses",
      "target": "000AYPA9AJ"
    },
    {
      "field": "uses",
      "target": "02AS45T3H9"
    },
    {
      "field": "uses",
      "target": "WD3RJZC5A7"
    },
    {
      "field": "uses",
      "target": "CCZA91NE5X"
    },
    {
      "field": "uses",
      "target": "3MSGFJM2NX"
    },
    {
      "field": "uses",
      "target": "K7N565NKCV"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/7VPJQNQV34.md",
  "msc": [
    "52A20",
    "52A05"
  ],
  "name": "Face of a convex set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7VP/7VPJQNQV34.md",
  "status": null,
  "tag": "7VPJQNQV34",
  "uri": "m2://7VPJQNQV34",
  "url": "https://tamnd.github.io/math2/wiki/face-of-a-convex-set-definition/",
  "writing": {
    "tag": "7VPJQNQV34",
    "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
      }
    ]
  }
}