{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $d \\ge 1$ and $r \\ge 2$ be integers, and let $X \\subseteq \\R^d$ be a finite set with at least $(r - 1)(d + 1) + 1$ points.\nThen $X$ can be [[partitioned|08QMKTWF9T]] into $r$ disjoint sets $X_1, \\dots, X_r$ whose [[convex hulls|0FQSDC2EQ2]] have a common point:\n$$\\operatorname{conv}(X_1) \\cap \\cdots \\cap \\operatorname{conv}(X_r) \\ne \\emptyset.$$\n\n## Remarks\n\nFor $r = 2$ the bound is $d + 2$, and the theorem is [[Radon's theorem|888R24ZGNY]].\n\nExample: with $d = 2$ and $r = 3$, any $7$ points in the plane can be split into three groups whose convex hulls share a point.\nFor instance, take the vertices of a large triangle, the vertices of a smaller triangle with the same centroid $c$, and the point $c$ itself.\nThe three groups formed by the large triangle, the small triangle and $\\set{c}$ all have $c$ in their convex hulls.\n\nThe bound $(r - 1)(d + 1) + 1$ is sharp: there are sets of $(r - 1)(d + 1)$ points in $\\R^d$ with no such partition.\n\nProof idea: Tverberg's original proof was a direct geometric argument.\nA shorter proof by Sarkaria lifts the points to a space of dimension $(r - 1)(d + 1)$ using tensor products with vectors that sum to $0$, and applies the [[colorful Carathéodory theorem|X5GS407VP8]] there.\n\n## Explanations\n\n### L1 human\n\nRadon's theorem splits enough points into two groups whose shapes overlap.\nTverberg's theorem does the same for any number $r$ of groups, all overlapping in one common point.\nIn the plane, seven points always split into three overlapping groups, and $3(r - 1) + 1$ points split into $r$ such groups.\n\n### L2 human\n\nA common point of a Tverberg partition is a kind of multidimensional median of the set, which links the theorem to the centerpoint theorem.\nThe topological Tverberg conjecture asked whether continuous maps of a simplex can replace convex hulls.\nIt holds when $r$ is a prime power and fails for some other values of $r$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "0FQSDC2EQ2"
    ],
    "kind": "statement",
    "msc": [
      "52A35",
      "52A37"
    ],
    "name": "Tverberg's theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q7857350",
      "wikipedia": "Tverberg's theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Tverberg%27s_theorem"
      }
    ],
    "tag": "V21XNZ1H0B",
    "uses": [
      "0XPR1E85W0",
      "0FQSDC2EQ2",
      "08QMKTWF9T",
      "888R24ZGNY",
      "X5GS407VP8"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/V21XNZ1H0B/",
  "links": [
    {
      "field": "about",
      "target": "0FQSDC2EQ2"
    },
    {
      "field": "uses",
      "target": "0XPR1E85W0"
    },
    {
      "field": "uses",
      "target": "0FQSDC2EQ2"
    },
    {
      "field": "uses",
      "target": "08QMKTWF9T"
    },
    {
      "field": "uses",
      "target": "888R24ZGNY"
    },
    {
      "field": "uses",
      "target": "X5GS407VP8"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/V21XNZ1H0B.md",
  "msc": [
    "52A35",
    "52A37"
  ],
  "name": "Tverberg's theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/V21/V21XNZ1H0B.md",
  "status": null,
  "tag": "V21XNZ1H0B",
  "uri": "m2://V21XNZ1H0B",
  "url": "https://tamnd.github.io/math2/wiki/tverbergs-theorem/",
  "writing": {
    "tag": "V21XNZ1H0B",
    "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
      }
    ]
  }
}