The maximum-width conjecture for hollow three-dimensional convex bodies

A hollow convex body is a three-dimensional convex body whose interior contains no lattice point. The width of a convex body is the minimum, over nonzero integer linear functionals, of the difference between its maximum and minimum values on the body. Let $w_c(3)$ denote the largest width of a hollow three-dimensional convex body. The tetrahedron constructed in the paper has width $2+\sqrt2$. Maximum-width conjecture. The tetrahedron in the construction is the convex $3$-body of largest width; equivalently,

$$w_c(3)=2+\sqrt2.$$

The construction establishes the lower bound $w_c(3)\geq 2+\sqrt2$; the conjecture asserts that this bound is sharp.