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  "body": "\n## Summary\n\nLet $K$ be a [[convex body|437FSG3419]] in $\\R^n$.\nIts projection body $\\Pi K$ is a convex body given by its [[support function|1H92CYA4ZC]].\nAt a unit vector $u$, this function is the volume of the shadow of $K$ on the hyperplane orthogonal to $u$.\nIt packs all the shadows of $K$ into one convex body, which is always symmetric about the origin.\nProjection bodies were introduced by Minkowski and are central in the study of affine isoperimetric inequalities.\n\n## Explanations\n\n### L1 human\n\nShine light on a solid from direction $u$ and measure the area of its shadow on a wall perpendicular to $u$.\nThe projection body is the convex shape whose width profile is given by these shadow areas.\nA cube of side $1$ has shadow area $\\abs{u_1} + \\abs{u_2} + \\abs{u_3}$ for a unit direction $u$, and its projection body is the cube $[-1, 1]^3$.\n",
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