A Riemannian metric on the two-sphere is called Zoll when all of its non-trivial geodesics are periodic and have the same period. The standard metric of the Euclidean sphere is an example but, perhaps surprisingly, it is not the only one. P. Funk and V. Guillemin studied the problem of deforming the canonical metric into new Zoll metrics with geodesics that are perturbations of the great circles of the Euclidean sphere. In a joint work with F. Codá and A. Neves, we studied the analogous problem in higher dimensional spheres, where now we seek to we perturb the family of totally geodesic great hyperspheres into a family of minimal hyperspheres for the new metric. In this talk, we will discuss some ingredients of the proof of this result, which include a careful analysis of Funk-Radon transforms.