  Let f(x) belong to L^p(R^n) and R>0. The transform is considered that
integrates the function f over (almost) all spheres of radius R in R^n. This
operator is known to be non-injective (as one can see by taking Fourier
transform). However, the counterexamples that can be easily constructed using
Bessel functions of the 1st kind, only belong to L^p if p>2n/(n-1). It has been
shown previously by S. Thangavelu that for p not exceeding the critical number
2n/(n-1), the transform is indeed injective.
  In this article, the support theorem is proven that strengthens this
injectivity result. Namely, if K is a convex bounded domain in R^n, the index p
is not above 2n/(n-1), and (almost) all the integrals of $f$ over spheres of
radius $R$ not intersecting K are equal to zero, then f is supported in the
closure of the domain K.
  In fact, convexity in this case is too strong a condition, and the result
holds for any what we call an R-convex domain.
