Gegenbauer-Chebyshev Integrals and Radon Transforms

The Radon transform $R$ assigns to a function $f$ on $R^n$  a collection 

of integrals of that function over   hyperplanes in $R^n$. Suppose 

that $Rf$ vanishes on  all hyperplanes that do not meet a fixed convex 

set. {\it Does it follow that $f$ is zero in the exterior of that set?}

I am planning to discuss new results related to this question and  the

corresponding injectivity problems. If time allows, some   projectively

equivalent modifications of $R$ will be considered.