Abstract: 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 $q$ and $p$, 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.