The Radon transform in two-dimensions is well studied. It maps a sufficiently nice function $f: \mathbb R^2 \to \mathbb R$ to its line integral along a certain line $L$, i.e.
\begin{align*} Rf(L) := \int_{L} f(x) \; \text{d}S. \end{align*}
The line $L$ can easily be parameterized by its angle $\alpha$ (say to the $x_1$-axis) and its displacement $s$ to the origin. We use same the notation and also write $Rf(\alpha,s)$.
It is a well-known result that if we know the Radon transform of a function $f$ for an open set of angles and all displacements, then $f$ can be uniquely determined. (By the Fourier-Slice theorem, the Fourier transform is given for a cone (indeed an open set in $\mathbb R^2$) and by the Paley-Wiener theorem knowing the Fourier transform on an open set is enough (Holomorphy argument)).
Now when it comes to the generalized Radon transform (on hyperplanes) however, I have not been able to find such "limited angle" result. Of course in the higher-dimensional case it does not make sense to talk about angles, but I think there should be some kind of analogy such as: "if one has enough freedom to spin the hyperplanes in $\mathbb R^n$, then uniqueness is given.
Literature and comments are greatly appreciated.Thank you.
