{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA convex polytope is the [[convex hull|0FQSDC2EQ2]] $P$ of a finite set of points of $\\R^d$.\nA hyperplane $H$ supports $P$ if $P$ lies in one of the two closed half-spaces bounded by $H$.\nA *face* of $P$ is $P$ itself, the empty set, or a set $P \\cap H$ for a supporting hyperplane $H$.\nThe *face lattice* of $P$ is the set of its faces, [[partially ordered|0QV2JS9RJ3]] by inclusion.\nTwo polytopes are *combinatorially equivalent* if their face lattices are isomorphic as partially ordered sets.\n*Polytope combinatorics* is the study of the properties of convex polytopes that depend only on their face lattice.\n\n## Remarks\n\nEach face is again a convex polytope, and its dimension is the dimension of its affine span; the empty face has dimension $-1$.\nFaces of dimension $0$, $1$ and $\\dim P - 1$ are called vertices, edges and facets.\n\nExample: a square and a trapezium in $\\R^2$ are combinatorially equivalent.\nEach has $4$ vertices and $4$ edges.\nMatch the vertices of one with the vertices of the other in cyclic order; this matching preserves which vertices lie on which edges.\n\nNon-example: a square and a triangle are not combinatorially equivalent, since their face lattices have $4$ and $3$ vertices.\n\nProperties such as the number of faces of each dimension, or whether every facet is a simplex, belong to polytope combinatorics; volume and edge lengths do not.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "879SBTVE7V",
    "kind": "definition",
    "msc": [
      "52B05"
    ],
    "name": "Polytope combinatorics",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Polyhedral_combinatorics"
      }
    ],
    "tag": "71VK7M83WS",
    "uses": [
      "0FQSDC2EQ2",
      "0QV2JS9RJ3"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/71VK7M83WS/",
  "links": [
    {
      "field": "defines",
      "target": "879SBTVE7V"
    },
    {
      "field": "uses",
      "target": "0FQSDC2EQ2"
    },
    {
      "field": "uses",
      "target": "0QV2JS9RJ3"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/71VK7M83WS.md",
  "msc": [
    "52B05"
  ],
  "name": "Polytope combinatorics",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/71V/71VK7M83WS.md",
  "status": null,
  "tag": "71VK7M83WS",
  "uri": "m2://71VK7M83WS",
  "url": "https://math2.tamnd.com/wiki/polytope-combinatorics-definition/",
  "writing": {
    "tag": "71VK7M83WS",
    "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
      }
    ]
  }
}