Sharp constant for a $k$-plane transform inequality

Abstract

The $k$-plane transform ${\mathcal{ℛ}}_k$ acting on test functions on ${ℝ}^d$ satisfies a dilation-invariant $L^p\to L^q$ inequality for some exponents $p,q$. We will make explicit some extremizers and the value of the best constant for any value of $k$ and $d$, solving the endpoint case of a conjecture of Baernstein and Loss. This extends their own result for $k=2$ and Christ’s result for $k=d-1$.