Seonglae ChoKakeya set
In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction. For instance, a disk of radius 1/2 in the Euclidean plane, or a ball of radius 1/2 in three-dimensional space, forms a Kakeya set. Much of the research in this area has studied the problem of how small such sets can be. Abram Besicovitch showed that there are Besicovitch sets of measure zero.
https://en.wikipedia.org/wiki/Kakeya_set
3d conjecture
Volume estimates for unions of convex sets, and the Kakeya set...
We study sets of $δ$ tubes in $\mathbb{R}^3$, with the property that not too many tubes can be contained inside a common convex set $V$. We show that the union of tubes from such a set must...
https://arxiv.org/abs/2502.17655
