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  "body": "\n## Definition\n\nLet $n \\ge 2$, and let $K \\subseteq \\R^n$ be a [[convex body|437FSG3419]].\nFor a nonzero vector $u$, let $K | u^\\perp$ be the [[orthogonal projection|0Q8HZTJC7G]] of $K$ onto the hyperplane $u^\\perp = \\set{x : x \\cdot u = 0}$.\nLet $\\mathrm{vol}_{n-1}$ be the $(n-1)$-dimensional volume in $u^\\perp$.\nThe *projection body* of $K$ is the convex body $\\Pi K$ whose [[support function|1H92CYA4ZC]] is\n$$h_{\\Pi K}(x) = \\abs{x} \\, \\mathrm{vol}_{n-1}\\bigl(K | x^\\perp\\bigr) \\quad \\text{for } x \\ne 0, \\qquad h_{\\Pi K}(0) = 0.$$\n\n## Remarks\n\nThat this function is a support function follows from Cauchy's projection formula $\\mathrm{vol}_{n-1}(K | u^\\perp) = \\frac{1}{2} \\int_{S^{n-1}} \\abs{u \\cdot v} \\, dS_K(v)$, where $S_K$ is the surface area measure of $K$.\nThe formula also shows that $\\Pi K$ is symmetric about the origin and is a [[zonoid|ZMZZ2QRFTZ]].\nExample: let $C = [0, 1]^n$.\nFor a unit vector $u$, the shadow of $C$ on $u^\\perp$ has volume $\\abs{u_1} + \\cdots + \\abs{u_n}$.\nThis is the support function of $[-1, 1]^n$ at $u$, so $\\Pi C = [-1, 1]^n$.\nNon-example: for $n \\ge 3$, every projection of a segment $K$ has $(n-1)$-volume $0$.\nSo the formula gives $h = 0$ and the set $\\set{0}$, which is not a convex body; this is why $K$ must have nonempty interior.\nFor the unit ball $B$, every shadow is an $(n-1)$-ball, so $\\Pi B = \\omega_{n-1} B$, with $\\omega_{n-1}$ the volume of the unit ball of $\\R^{n-1}$.\n",
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