Universal convex coverings

Abstract

In every dimension $d\ge 1$, we establish the existence of a constant $v_d>0$ and of a subset ${\mathcal{U}}_d$ of ${\mathbb{R}}^d$ such that the following holds: $\mathcal{C}+{\mathcal{U}}_d={\mathbb{R}}^d$ for every convex set $\mathcal{C}\subset {\mathbb{R}}^d$ of volume at least $v_d$ and ${\mathcal{U}}_d$ contains at most $\log (r{)}^{d-1}{r}^d$ points at distance at most $r$ from the origin, for every large $r$.

8 pages