On lattice illumination of smooth convex bodies

The illumination conjecture is a classical open problem in convex and discrete geometry, asserting that every compact convex body~$K$ in ${\mathbb{R}}^n$ can be illuminated by a set of no more than $2^n$ points. If $K$ has smooth boundary, it is known that $n+1$ points are necessary and sufficient. We consider an effective variant of the illumination problem for bodies with smooth boundary, where the illuminating set is restricted to points of a lattice and prove the existence of such a set close to $K$ with an explicit bound on the maximal distance. We produce improved bounds on this distance for certain classes of lattices, exhibiting additional symmetry or near-orthogonality properties. Our approach is based on the geometry of numbers.