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  "body": "\n## Problem\n\nLet $n \\ge 2$, and let $S^{n-1}$ be the unit [[sphere|T62369RFH6]] of $\\R^n$.\nLet $\\mathcal{H}^{n-1}$ be the $(n-1)$-dimensional [[Hausdorff measure|Q1TGZK8RN6]].\nLet $K \\subseteq \\R^n$ be a [[convex body|437FSG3419]] and $\\omega \\subseteq S^{n-1}$ a Borel set.\nLet $N_K(\\omega)$ be the set of boundary points of $K$ with an outer unit normal in $\\omega$.\nThe *surface area measure* of $K$ is the Borel measure $S_K(\\omega) = \\mathcal{H}^{n-1}(N_K(\\omega))$ on $S^{n-1}$.\nLet $\\mu$ be a finite Borel measure on $S^{n-1}$.\nFind a convex body $K$ with $S_K = \\mu$, and decide when such a body exists and whether it is unique.\n\n## Remarks\n\nThe problem is solved.\nA convex body $K$ with $S_K = \\mu$ exists if and only if $\\mu$ is not concentrated on a great subsphere of $S^{n-1}$ and $\\int_{S^{n-1}} u \\, d\\mu(u) = 0$.\nThe body is then unique up to translation.\nMinkowski solved the case of polytopes.\nLet $u_1, \\ldots, u_m$ be distinct unit vectors spanning $\\R^n$, and let $a_1, \\ldots, a_m \u003e 0$ with $\\sum_i a_i u_i = 0$.\nThen there is a polytope, unique up to translation, whose facets have outer normals $u_i$ and areas $a_i$.\nThe general case is due to Minkowski, Alexandrov, and Fenchel and Jessen.\nWhen $\\mu$ has a positive density $f$, the problem asks for a body whose [[Gaussian curvature|SQVQ5DQT6W]] at the point with outer normal $u$ is $1 / f(u)$.\nThis is a [[Monge–Ampère equation|14AKAC0S0T]] on the sphere.\nThe regularity of its solutions was studied by Pogorelov and by Cheng and Yau.\nExample: the uniform measure $\\mu = c \\, \\sigma$ on $S^{n-1}$, with $\\sigma$ the surface measure and $c \u003e 0$, is the surface area measure of a ball of radius $c^{1/(n-1)}$.\n\n## Explanations\n\n### L1 human\n\nSuppose you know, for each direction, how much of the surface of a convex shape faces that direction.\nCan you rebuild the shape?\nMinkowski's answer is yes, up to sliding it around, provided the data are balanced: the outward pulls of all the pieces of surface must cancel.\nFor a polyhedron, the data are the directions and areas of its faces.\n",
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      "first_stated": "1897",
      "posed_by": "Hermann Minkowski"
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