Uniqueness of extremizers for an endpoint inequality of the $k$-plane transform

The $k$-plane transform is a bounded operator from $\text{lp}$ to $L^q$ of the Grassmann manifold of all affine $k$-planes in ${\mathbb{R}}^n$ for certain exponents depending on $k$ and $n$. In the endpoint case $q=n+1$, we identify all extremizers of the associated inequality for the general $k$-plane transform.