{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $d \\ge 1$ and $N \\ge 1$, and let $S_1, \\dots, S_N$ be subsets of $\\R^d$.\nLet $S_1 + \\cdots + S_N$ be their [[Minkowski sum|SXS6D87PHE]] and let $\\operatorname{conv}$ denote the [[convex hull|0FQSDC2EQ2]].\nThen every point $x \\in \\operatorname{conv}(S_1 + \\cdots + S_N)$ can be written as $x = x_1 + \\cdots + x_N$ with $x_i \\in \\operatorname{conv}(S_i)$ for each $i$, and with $x_i \\in S_i$ for all but at most $d$ indices $i$.\n\n## Remarks\n\nFirst, $\\operatorname{conv}(S_1 + \\cdots + S_N) = \\operatorname{conv}(S_1) + \\cdots + \\operatorname{conv}(S_N)$, so $x = \\sum_i \\sum_j \\lambda_{ij} y_{ij}$ with $y_{ij} \\in S_i$, $\\lambda_{ij} \\ge 0$ and $\\sum_j \\lambda_{ij} = 1$ for each $i$.\nThen the point $(x, 1, \\dots, 1) \\in \\R^{d+N}$ is a nonnegative combination of the vectors $(y_{ij}, e_i)$, where $e_i$ is the $i$-th unit vector of $\\R^N$.\nBy the conic form of [[Carathéodory's theorem|0201Q4P2B5]], at most $d + N$ of these vectors are needed.\nEach index $i$ needs at least one of them, so at most $d$ indices use two or more, and the other indices use a single point of $S_i$.\nExample: $d = 1$ and $S_i = \\set{0, 1}$ for all $i$.\nThe point $x = 2.5$ of $\\operatorname{conv}(S_1 + S_2 + S_3) = [0, 3]$ is $1 + 1 + 0.5$, and only one summand is outside its $S_i$.\n\n## Explanations\n\n### L1 human\n\nAdding many sets together makes the sum look almost convex, even if the sets themselves are far from convex.\nThe lemma makes this precise: a point of the convex hull of the sum can be reached by taking actual points of all but $d$ of the sets.\nThe number $d$ is the dimension, and it does not grow with the number of sets.\n\n### L2 human\n\nA quantitative consequence, the Shapley–Folkman theorem, bounds the distance between $\\operatorname{conv}(S_1 + \\cdots + S_N)$ and $S_1 + \\cdots + S_N$ in terms of $d$ and the sizes of the $S_i$, independently of $N$.\nIn mathematical economics this explains why equilibria nearly exist in large markets with nonconvex preferences.\nIn optimization it bounds the duality gap of separable nonconvex problems with many terms.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "SXS6D87PHE",
      "0FQSDC2EQ2"
    ],
    "kind": "statement",
    "msc": [
      "52A20",
      "91B50"
    ],
    "name": "Shapley–Folkman lemma",
    "role": "lemma",
    "same_as": {
      "wikidata": "Q3788045",
      "wikipedia": "Shapley–Folkman lemma"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Shapley–Folkman_lemma"
      }
    ],
    "tag": "CMGKQNMY77",
    "uses": [
      "0FQSDC2EQ2",
      "SXS6D87PHE"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/CMGKQNMY77/",
  "links": [
    {
      "field": "about",
      "target": "SXS6D87PHE"
    },
    {
      "field": "about",
      "target": "0FQSDC2EQ2"
    },
    {
      "field": "uses",
      "target": "0FQSDC2EQ2"
    },
    {
      "field": "uses",
      "target": "SXS6D87PHE"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/CMGKQNMY77.md",
  "msc": [
    "52A20",
    "91B50"
  ],
  "name": "Shapley–Folkman lemma",
  "prov": null,
  "role": "lemma",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CMG/CMGKQNMY77.md",
  "status": null,
  "tag": "CMGKQNMY77",
  "uri": "m2://CMGKQNMY77",
  "url": "https://tamnd.github.io/math2/wiki/shapley-folkman-lemma/",
  "writing": {
    "tag": "CMGKQNMY77",
    "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
      }
    ]
  }
}