Stability for the adjoint Finner inequality and improved lower bounds on the $k$-plane transform

Abstract

We establish stability properties of near-extremisers for the Loomis--Whitney inequality and its adjoint, presenting our results in the general framework of Finner. In particular, we show that a function nearly achieves equality in the adjoint Finner inequality if and only if it is near to being an indicator function of a cartesian product. As an application, we obtain new lower bounds for the classical $k$-plane transform on $L^p(\mathbb{R}^d)$, modestly improving those obtained recently by Tao and the first author. We also present an alternative approach to such improvements based on stability in the setting of the Brascamp--Lieb inequality and the theory of fusion frames. We conclude with an interpretation of our $k$-plane transform estimates as adjoint forms of the continuous Brascamp--Lieb inequalities.