{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $P$ be a [[convex polytope|300K6CB1KH]] of dimension $d$, and let $k \\ge 1$ be an integer.\n$P$ is *$k$-neighborly* if for every set $S$ of $k$ [[vertices|R7V3S9V891]] of $P$, the convex hull of $S$ is a [[face|KY7SB854VH]] of $P$.\n$P$ is *neighborly* if it is $\\floor{d/2}$-neighborly.\n\n## Remarks\n\nEvery polytope is $1$-neighborly, so in dimensions $2$ and $3$ every polytope is neighborly.\nA $2$-neighborly polytope is one whose graph is complete: every two vertices are joined by an edge.\nIf $P$ is $k$-neighborly for some $k \u003e \\floor{d/2}$, then $P$ is a simplex.\n\nExample: the cyclic polytope $C(n, 4)$ for $n \\ge 5$ is $2$-neighborly.\nFor $i \u003c j$, write $p(t) = (t - t_i)^2 (t - t_j)^2 = a_0 + a_1 t + \\dots + a_4 t^4$.\nThe affine function $\\ell(y) = a_0 + a_1 y_1 + \\dots + a_4 y_4$ satisfies $\\ell(x(t)) = p(t) \\ge 0$.\nIt vanishes only at the vertices with parameters $t_i$ and $t_j$, so these two vertices span an edge.\n\nNon-example: the $4$-cube $[0,1]^4$ is not neighborly, since the opposite vertices $0$ and $(1,1,1,1)$ are not joined by an edge.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "9V4PR1RVXW",
    "kind": "definition",
    "msc": [
      "52B05",
      "52B11",
      "52B12"
    ],
    "name": "Neighborly polytope",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Neighborly_polytope"
      }
    ],
    "tag": "TN1X6FNT87",
    "uses": [
      "300K6CB1KH",
      "R7V3S9V891",
      "KY7SB854VH"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/TN1X6FNT87/",
  "links": [
    {
      "field": "defines",
      "target": "9V4PR1RVXW"
    },
    {
      "field": "uses",
      "target": "300K6CB1KH"
    },
    {
      "field": "uses",
      "target": "R7V3S9V891"
    },
    {
      "field": "uses",
      "target": "KY7SB854VH"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/TN1X6FNT87.md",
  "msc": [
    "52B05",
    "52B11",
    "52B12"
  ],
  "name": "Neighborly polytope",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TN1/TN1X6FNT87.md",
  "status": null,
  "tag": "TN1X6FNT87",
  "uri": "m2://TN1X6FNT87",
  "url": "https://tamnd.github.io/math2/wiki/neighborly-polytope-definition/",
  "writing": {
    "tag": "TN1X6FNT87",
    "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
      }
    ]
  }
}