2 problems

Universal-cover-of-balls conjecture. There exist positive dimensional constants $\delta_n$, $C_n$, and $R_n$ such that, for every $p\in M$,

Macroscopic curvature and codimension 2 width. There exist positive dimensional constants $\delta_n$, $C_n$, and $R_n$ such that, if every ball of radius $R_n$ in $M$ has volume at…