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  "body": "\n## Definition\n\nLet $n \\ge 2$, and let $K \\subseteq \\R^n$ be a [[convex body|437FSG3419]] with the origin in its interior.\nLet $S^{n-1}$ be the unit [[sphere|T62369RFH6]] of $\\R^n$, and for $u \\in S^{n-1}$ let $u^\\perp = \\set{x : x \\cdot u = 0}$.\nWrite $\\mathrm{vol}_{n-1}$ for the $(n-1)$-dimensional volume in the hyperplane $u^\\perp$.\nThe *intersection body* of $K$ is the set\n$$IK = \\set{r u : u \\in S^{n-1}, \\ 0 \\le r \\le \\mathrm{vol}_{n-1}(K \\cap u^\\perp)}.$$\nSo $IK$ is star-shaped about the origin, and its radial function is $\\rho_{IK}(u) = \\mathrm{vol}_{n-1}(K \\cap u^\\perp)$.\n\n## Remarks\n\nExample: let $B$ be the closed unit ball of $\\R^n$ and $\\omega_{n-1}$ the volume of the unit ball of $\\R^{n-1}$.\nEvery section $B \\cap u^\\perp$ is a unit ball of dimension $n - 1$, so $\\rho_{IB} = \\omega_{n-1}$ and $IB = \\omega_{n-1} B$.\nNon-example: let the origin lie outside $K$.\nThen some hyperplane $u^\\perp$ misses $K$, and the radial function vanishes in the direction $u$.\nSo the construction no longer gives a body with the origin in its interior, which is why the definition requires the origin to be an interior point of $K$.\nBusemann's theorem says that $IK$ is convex when $K$ is convex and symmetric about the origin.\nThe definition is often stated more generally for star bodies, with the same formula for the radial function.\n",
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