3 problems

Let $r_n$ be any consistent estimator of the reach, and let $(\beta_n)$ be any sequence such that $\beta_n\to 0$. Arbitrarily slow convergence conjecture. It is possible to find a…

Let $S$ be a subset of Euclidean space, and let $r>0$. The set $S$ is $r$-convex when it equals its $r$-convex hull, and it satisfies the $r$-rolling condition when every boundary…

Let $d$ be a positive integer, let $[0,1]^d$ be the unit cube, and let $\mu>0$ be given. Consider the class of all convex polytopes contained in $[0,1]^d$ whose volume is greater t…