Cube always hitting a lattice point

Problem's Description

Is there a universal constant $c$ such that all translation and rotations of a cube $Q \subset \mathbb{R}^n$ with side-length $c$ always contains a point from the $\mathbb{Z}^n$?

The problem is that in high dimensions the lattice $\mathbb{Z}^n$ has gigantic holes and it is not so easy to rule out the possibility of a suitably rotated large cube hiding there. Henk-Tsintifas prove that $c=\sqrt{2}$ when $n=2$ or $n=3$.

1. Article Lattice point coverings

   • Martin Henk,
   • George Tsintsifas

   Chinese Advances in Mathematics, 2007