The Kakeya problem for codimension-two planes in $\mathbb{R}^n$

Abstract

We study the $(n-2)$-plane Kakeya problem in $\mathbb{R}^n$ ($n\geq4$) subject to convex Wolff axioms. In other words, we study finite families of $\delta$-neighborhoods of $(n-2)$-plane segments ($\delta$-flakes) in $\mathbb{R}^n$, with the property that not too many of them can be contained inside a common convex set. Our main result is that the union of $\delta$-flakes from such a set must have almost maximal volume. This can be viewed as a higher-dimensional generalization of Wang and Zahl's work for $\delta$-tubes in $\mathbb{R}^3$. The latter implies the Kakeya set conjecture in $\mathbb{R}^3$. As a consequence, we resolve Rogers'$2$-plane Kakeya problem in $\mathbb{R}^4$ subject to certain curvature conditions on two-dimensional families of direction subspaces in the Grassmannian $\mathrm{Gr}(4,2)$.