7 problems

Fix $n\geq 2$ and $0<\delta\ll 1$. Let $\Omega$ be a maximal $\delta$-separated subset of directions on $S^{n-1}$, and for each $\omega\in\Omega$ let $T_\omega$ be a…

Let $0<\delta\leq\nu<1$. Three families $\mathbb{T}_{1},\mathbb{T}_{2},\mathbb{T}_{3}$ of $\delta$-tubes are $\nu$-disjoint when their orientations lie in spherical patches of diam…

Let $0<\delta<1$. A $\delta$-tube is a tube of length $1$ and cross-sectional diameter $\delta$, and a family $\mathbb{T}$ of $\delta$-tubes is $\delta$-separated when distinct tub…

Fix $d\geq2$. For each $1/R$-cap $\theta$ on $S^{d-1}$, let $\theta^*$ be the corresponding radius-one, length-$R$ tube, and let $x_\theta$ be arbitrary. Kakeya $L^p$ conjecture. F…

Let $0<\varepsilon<1$, and let $\mathcal{K}^{\ast}(\otimes_1L^{\infty}\rightarrow L^{\frac{3}{2}};\varepsilon)$ be the statement that for every $0<\delta<1$ and every family…

Let $E$ be an arbitrary subset of $\mathbb{R}^n$, and let $D$ be the set of directions in which $E$ contains a line segment. General Kakeya conjecture. If $D$ is non-empty, then ……

X-ray endpoint conjecture. For every $\varepsilon>0$, there is a constant $C_\varepsilon<\infty$ such that