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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 that is symmetric about the origin, that is, $K = -K$.\nLet $K^\\circ = \\set{y \\in \\R^n : x \\cdot y \\le 1 \\text{ for all } x \\in K}$ be its polar body, where $x \\cdot y$ is the [[dot product|0S52TNNTH9]].\nLet $\\operatorname{vol}$ be [[Lebesgue measure|2Z2QVWZ262]] and $\\kappa_n$ the volume of the closed unit ball $B$ of $\\R^n$.\nThen\n$$\\operatorname{vol}(K) \\operatorname{vol}(K^\\circ) \\le \\kappa_n^2,$$\nwith equality if and only if $K$ is an ellipsoid centered at the origin.\n\n## Remarks\n\nThe product $\\operatorname{vol}(K) \\operatorname{vol}(K^\\circ)$ is the Mahler volume of $K$; it does not change when $K$ is replaced by $AK$ for an invertible linear map $A$, because $(AK)^\\circ = (A^T)^{-1} K^\\circ$.\nA standard proof uses Steiner symmetrization: symmetrizing $K$ in a hyperplane does not decrease the Mahler volume, and suitable sequences of symmetrizations converge to a ball.\nThe inequality holds without symmetry if the origin is replaced by the Santaló point of $K$, the unique interior point $z$ that minimizes $\\operatorname{vol}((K - z)^\\circ)$.\n\nExample: $K = [-1, 1]^2$ has $K^\\circ = \\set{(y_1, y_2) : \\abs{y_1} + \\abs{y_2} \\le 1}$, of area $2$.\nSo the Mahler volume is $4 \\cdot 2 = 8$, which is less than $\\pi^2 \\approx 9.87$.\n\nSome centering is needed: for $K = [-1, 1]^2 - (1 - \\varepsilon, 0)$, the origin is at distance $\\varepsilon$ from the boundary, and $\\operatorname{vol}(K^\\circ) \\to \\infty$ as $\\varepsilon \\to 0$.\n\n## Explanations\n\n### L1 human\n\nThe polar body $K^\\circ$ is a kind of reciprocal of $K$: when $K$ gets bigger, $K^\\circ$ gets smaller.\nThe product of their volumes is unchanged by stretching.\nThe Blaschke–Santaló inequality says that this product is largest for balls and ellipsoids.\n\n### L2 human\n\nThe inequality is a basic tool of asymptotic convex geometry and has functional forms for log-concave functions.\nThe reverse question, how small $\\operatorname{vol}(K) \\operatorname{vol}(K^\\circ)$ can be, is the [[Mahler conjecture|FFZHNFJCZS]].\nFor symmetric $K$, the Bourgain–Milman inequality gives a lower bound $c^n \\kappa_n^2$ with an absolute constant $c \u003e 0$.\n",
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        "url": "https://en.wikipedia.org/wiki/Mahler_volume"
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  "msc": [
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  "name": "Blaschke–Santaló inequality",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/MNJ/MNJ030R4CA.md",
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