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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $K \\subseteq \\R^n$ be a compact [[convex body|437FSG3419]] with nonempty interior and $K = -K$.\nLet $K^\\circ = \\set{y \\in \\R^n : x \\cdot y \\le 1 \\text{ for all } x \\in K}$ be its polar body, with $x \\cdot y$ the [[dot product|0S52TNNTH9]], and let $\\operatorname{vol}$ be [[Lebesgue measure|2Z2QVWZ262]].\nThe conjecture states that\n$$\\operatorname{vol}(K) \\operatorname{vol}(K^\\circ) \\ge \\frac{4^n}{n!},$$\nwhich is the value for the cube $[-1, 1]^n$.\n\n## Remarks\n\nFor the cube, $\\operatorname{vol}([-1,1]^n) = 2^n$ and its polar is the cross-polytope of volume $2^n / n!$, which gives $4^n / n!$.\nThe conjecture is proved for $n = 2$, by Mahler, and for $n = 3$, by Iriyama and Shibata.\nIt is known for some classes of bodies, such as unconditional bodies and zonoids.\nIn general the Bourgain–Milman inequality gives $\\operatorname{vol}(K) \\operatorname{vol}(K^\\circ) \\ge c^n \\frac{4^n}{n!}$ for an absolute constant $c \u003e 0$, and Kuperberg gave an explicit such constant.\nThe upper bound is the [[Blaschke–Santaló inequality|MNJ030R4CA]].\nThere is also a version for convex bodies that are not symmetric, in which the simplex is the conjectured minimizer.\n\n## Explanations\n\n### L1 human\n\nMultiply the volume of a symmetric convex shape by the volume of its polar shape.\nThe result does not change when the shape is stretched.\nThe largest possible product is reached by balls.\nMahler conjectured that the smallest possible product is reached by the cube, and this is still open in dimension four and higher.\n",
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    "history": {
      "first_stated": "1939",
      "posed_by": "Kurt Mahler"
    },
    "kind": "conjecture",
    "msc": [
      "52A40",
      "52A20"
    ],
    "name": "Mahler conjecture",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Mahler_volume"
      }
    ],
    "tag": "FFZHNFJCZS",
    "uses": [
      "437FSG3419",
      "2Z2QVWZ262",
      "0S52TNNTH9"
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  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/FFZHNFJCZS/",
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  "markdown": "https://tamnd.github.io/math2/node/FFZHNFJCZS.md",
  "msc": [
    "52A40",
    "52A20"
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  "name": "Mahler conjecture",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FFZ/FFZHNFJCZS.md",
  "status": null,
  "tag": "FFZHNFJCZS",
  "uri": "m2://FFZHNFJCZS",
  "url": "https://tamnd.github.io/math2/wiki/mahler-conjecture/",
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    "tag": "FFZHNFJCZS",
    "kind": "conjecture",
    "level": "W0",
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