Let $S$ be a hypersphere of radius $r$ in $\mathbb{R}^n$ whose centre may or may not be a lattice point (i.e. a point in $\mathbb{Z}^n$). Can you give any upper bound to the volume of the smallest lattice polyhedron (i.e. a polyhedron with vertices in $\mathbb{Z}^n$) which contains $S$?
My thoughts: Clealy $S$ is contained in a concentric hypercube of side length $2r$ - only problem is this hypercube need not be a lattice polyhedron. But I was thinking that maybe there is some lattice polytope containing this hypercube which lies in a concentric hypercube of side length $2r + \epsilon$ - but I am not able to find such an $\epsilon$?