On lattice illumination of smooth convex bodies

TL;DR

This paper investigates a lattice-based variant of the classical illumination problem for smooth convex bodies, providing explicit bounds and improved results for special lattice classes using geometry of numbers.

Contribution

It introduces a lattice-restricted illumination problem, proves the existence of near-boundary lattice points illuminating smooth convex bodies, and improves bounds for symmetric or near-orthogonal lattices.

Findings

Existence of lattice points close to smooth convex bodies with explicit bounds.

Improved bounds for bodies with symmetric or near-orthogonal lattices.

Application of geometry of numbers to convex illumination problems.

Abstract

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.