The Radon transform is an integral transform with applications in mathematics, tomography, and medicine. The k-plane transform is an integral transform is that maps a function to its integral over all k- dimensional planes. When k=n-1, the k-plane transform and the Radon transform coincide.

The k-plane transform is a bounded operator from L^p of Euclidean space to L^q of the Grassman manifold of all affine k-planes. Extremizers have been determined for certain values of p and q, but most remain open. The focus will be showing that when q and the reciprocal of p-1 are both integers, any extremizer is smooth function. This involves analysis of a nonlinear Euler- Lagrange equation.