The generalized Radon transform maps a function $f \in L^1(\mathbb R^n)$, usually interpreted as a density of an object, to its integral value over an $(n-1)$-dimensional affine subspace.
To be more precise, if $H$ is an affine hyperplane parameterized via $H = \{ x \in \mathbb R^n \; : \; \langle \omega, x \rangle = s \}$, then \begin{align*} (Rf)(H) = (Rf)(\omega, s) = \int_{H} f \; \text{d}S. \end{align*}
It is a well-known result (cf Helgason Sigurdur: Integral geometry and Radon transforms) that knowing the Radon transform for all affine hyperplanes is sufficient to determine $f$.
My question is whether knowing the Radon transform for really all affine hyperplanes is really necessary to determine $f$ uniquely.
For example for $n=2$ the hyperplanes are clearly lines and the transform is bettern known as X-Ray transform. For this transform we know that an infinite number of directions of "rays" (plus all possible displacements from the origin) is already sufficient, i.e. for $n=2$ the condition is not necessary.
How does this behave in the more general cases, i.e. $n>2$ ?
