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

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.