Helgason support theorem

Remarks

The decay hypothesis cannot be dropped. In $\R^2 = \C$, let $k \ge 2$ be an integer and let $f$ be continuous with $f(z) = z^{-k}$ for $\abs{z} \ge 1$. Take a line $z = z_0 + t w$, $t \in \R$, that misses the closed unit disk. The function $t \mapsto (z_0 + tw)^{-k}$ has its only pole off the real axis, and it decays like $\abs{t}^{-k}$. Closing the contour in the half-plane without the pole gives integral $0$. So $R f$ 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.

The convexity of $K$ is also needed. The 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.

Going deeper

For a student

The theorem gives the exterior problem of tomography its uniqueness: a body is determined outside a convex obstacle by the lines that avoid the obstacle. It is a sharper form of injectivity of the Radon transform, local in the set of hyperplanes. The example with $z^{-k}$ shows that rapid decrease is the right hypothesis. The theorem is due to Helgason and has analogues for Radon transforms on symmetric spaces.