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  "body": "\n## Statement\n\nLet $n \\ge 1$, and let $K \\subseteq \\R^n$ be a [[convex body|437FSG3419]].\nThen there are $\\lambda \\in (0, 1)$ and points $x_1, \\ldots, x_{2^n} \\in \\R^n$ with\n$$K \\subseteq \\bigcup_{i=1}^{2^n} (x_i + \\lambda K),$$\nthat is, $K$ can be covered by $2^n$ smaller [[homothetic|0ZG1PNEP35]] copies of itself.\nMoreover, $2^n$ copies are needed only when $K$ is a parallelotope.\n\n## Remarks\n\nThe conjecture is open for $n \\ge 3$; Levi proved the case $n = 2$.\nThe cube $[0, 1]^n$ shows that $2^n$ cannot be lowered.\nA translate of $\\lambda [0, 1]^n$ with $\\lambda \u003c 1$ has all side lengths below $1$, so it contains at most one of the $2^n$ vertices.\nGeneral upper bounds of exponential order, about $4^n$ times a polynomial factor in $n$, are known.\nBoltyanski showed that the conjecture is equivalent to a statement about illuminating the boundary of $K$ by $2^n$ directions.\n\n## Explanations\n\n### L1 human\n\nThe conjecture says that any convex shape in $n$-dimensional space can be covered by $2^n$ slightly shrunk copies of itself.\nIn the plane four copies are enough, and a square really needs four, one for each corner.\nIn space it predicts eight copies, but this is still unproved.\n",
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    "history": {
      "first_stated": "1957",
      "posed_by": "Hugo Hadwiger"
    },
    "kind": "conjecture",
    "msc": [
      "52C17",
      "52A20"
    ],
    "name": "Hadwiger covering conjecture",
    "same_as": {
      "wikidata": "Q5638115",
      "wikipedia": "Hadwiger conjecture (combinatorial geometry)"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hadwiger_conjecture_(combinatorial_geometry)"
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    "tag": "SVP75YY386",
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  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/SVP75YY386/",
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  "markdown": "https://tamnd.github.io/math2/node/SVP75YY386.md",
  "msc": [
    "52C17",
    "52A20"
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  "name": "Hadwiger covering conjecture",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SVP/SVP75YY386.md",
  "status": null,
  "tag": "SVP75YY386",
  "uri": "m2://SVP75YY386",
  "url": "https://tamnd.github.io/math2/wiki/hadwiger-covering-conjecture/",
  "writing": {
    "tag": "SVP75YY386",
    "kind": "conjecture",
    "level": "W0",
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