{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "V21XNZ1H0B"
    }
  ],
  "body": "\n## Statement\n\nLet $d \\ge 1$ and let $C_1, \\dots, C_{d+1}$ be subsets of $\\R^d$.\nLet $x \\in \\R^d$ be a point that lies in the [[convex hull|0FQSDC2EQ2]] of each $C_i$, for $i = 1, \\dots, d + 1$.\nThen there are points $c_1 \\in C_1, \\dots, c_{d+1} \\in C_{d+1}$ such that\n$$x \\in \\operatorname{conv}\\set{c_1, \\dots, c_{d+1}}.$$\n\n## Remarks\n\nProof idea: by [[Carathéodory's theorem|0201Q4P2B5]] one may assume each $C_i$ is finite.\nAmong all choices of $c_1 \\in C_1, \\dots, c_{d+1} \\in C_{d+1}$, pick one for which the distance from $x$ to $\\operatorname{conv}\\set{c_1, \\dots, c_{d+1}}$ is smallest.\nIf this distance were positive, the nearest point $y$ would lie in the hull of at most $d$ of the $c_j$, so some $c_i$ could be dropped.\nThe hyperplane through $y$ perpendicular to $x - y$ has all the $c_j$ on one side.\nSince $x \\in \\operatorname{conv}(C_i)$, some point of $C_i$ lies strictly on the side of $x$.\nReplacing $c_i$ by that point brings the hull closer to $x$, a contradiction.\n\nWhen all the $C_i$ are the same set $C$, the theorem is Carathéodory's theorem for $C$.\n\nExample: in the plane, let $C_1$, $C_2$, $C_3$ be red, green and blue point sets, each with the origin in its convex hull.\nThen some triangle with one red, one green and one blue vertex contains the origin.\n\n## Explanations\n\n### L1 human\n\nColour points with $d + 1$ colours, and suppose that the points of each colour surround a given point $x$.\nThen you can pick one point of each colour so that these $d + 1$ points already surround $x$.\nIn the plane: three colour classes each surrounding $x$ give a rainbow triangle around $x$.\n\n### L2 human\n\nThe theorem is due to Imre Bárány.\nIt is the key tool in Sarkaria's proof of Tverberg's theorem, and Bárány used it to prove the first selection lemma on simplices that contain a common point.\nIt also has a colourful counterpart for Helly's theorem.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "0FQSDC2EQ2"
    ],
    "kind": "statement",
    "msc": [
      "52A35"
    ],
    "name": "Colorful Carathéodory theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_(convex_hull)"
      }
    ],
    "tag": "X5GS407VP8",
    "uses": [
      "0XPR1E85W0",
      "0FQSDC2EQ2",
      "0201Q4P2B5"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/X5GS407VP8/",
  "links": [
    {
      "field": "about",
      "target": "0FQSDC2EQ2"
    },
    {
      "field": "uses",
      "target": "0XPR1E85W0"
    },
    {
      "field": "uses",
      "target": "0FQSDC2EQ2"
    },
    {
      "field": "uses",
      "target": "0201Q4P2B5"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/X5GS407VP8.md",
  "msc": [
    "52A35"
  ],
  "name": "Colorful Carathéodory theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/X5G/X5GS407VP8.md",
  "status": null,
  "tag": "X5GS407VP8",
  "uri": "m2://X5GS407VP8",
  "url": "https://tamnd.github.io/math2/wiki/colorful-caratheodory-theorem/",
  "writing": {
    "tag": "X5GS407VP8",
    "kind": "statement",
    "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
      }
    ]
  }
}