Volumes of balls in Riemannian manifolds and Uryson width

If $(M^n, g)$ is a closed Riemannian manifold where every unit ball has volume at most $\epsilon_n$ (a sufficiently small constant), then the $(n-1)$-dimensional Uryson width of $(M^n, g)$ is at most 1.Comment: 26 page

A short proof of the multilinear Kakeya inequality

We give a short proof of a slightly weaker version of the multilinear Kakeya inequality proven by Bennett, Carbery, and Tao.Comment: 7 pages, minor revision, accepted for publication in the Mathematical Proceedings of the Cambridge Philosophical Societ