2025/01/17 by Fukshansky, Lenny
#11H06 #52A05 #52C07 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2501.10570
The illumination conjecture is a classical open problem in convex and discrete geometry, asserting that every compact convex body~K in \mathbb Rn can be illuminated by a set of no more than 2n 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.