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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $f \\colon \\R^n \\to \\C$ be continuous and rapidly decreasing, that is, $\\abs{x}^k f(x)$ is bounded on $\\R^n$ for every integer $k \\ge 0$.\nLet $K \\subseteq \\R^n$ be a compact convex set.\nAssume that the [[Radon transform|VJNZJ352PC]] of $f$ vanishes on every hyperplane that does not meet $K$.\nThat is, $R f(\\theta, s) = 0$ whenever the hyperplane $x \\cdot \\theta = s$ is disjoint from $K$.\nThen $f(x) = 0$ for every $x \\notin K$.\n\n## Remarks\n\nThe decay hypothesis cannot be dropped.\nIn $\\R^2 = \\C$, let $k \\ge 2$ be an integer and let $f$ be continuous with $f(z) = z^{-k}$ for $\\abs{z} \\ge 1$.\nTake a line $z = z_0 + t w$, $t \\in \\R$, that misses the closed unit disk.\nThe function $t \\mapsto (z_0 + tw)^{-k}$ has its only pole off the real axis, and it decays like $\\abs{t}^{-k}$.\nClosing the contour in the half-plane without the pole gives integral $0$.\nSo $R f$ vanishes on all lines missing the unit disk, while $f \\ne 0$ outside it; here $\\abs{x}^k f(x)$ is bounded but $\\abs{x}^{k+1} f(x)$ is not.\n\nThe convexity of $K$ is also needed.\nThe hyperplanes that miss a compact set are those that miss its convex hull, so the data can only locate the support up to the convex hull.\n\n## Explanations\n\n### L1 human\n\nSuppose every X-ray that misses a certain convex region sees nothing.\nThe theorem says that, under a mild decay condition, there is indeed nothing outside that region.\nIt is not obvious: a ray that misses the region could in principle pass through material whose contributions cancel.\n\n### L2 human\n\nThe theorem gives the exterior problem of tomography its uniqueness: a body is determined outside a convex obstacle by the lines that avoid the obstacle.\nIt is a sharper form of injectivity of the [[Radon transform|VJNZJ352PC]], local in the set of hyperplanes.\nThe example with $z^{-k}$ shows that rapid decrease is the right hypothesis.\nThe theorem is due to Helgason and has analogues for Radon transforms on symmetric spaces.\n",
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