A criterion for finite lattice coverings

Abstract

For a centrally symmetric convex \(K \subset E^2 \) and a covering lattice L for K, a lattice polygon P is called a covering polygon, if \(P \subset (L \cap P) + K\). We prove that P is a covering polygon, if and only if its boundary bd(P) is covered by (L ∩ P) + K. Further we show that this characterization is false for non-symmetric planar convex bodies and in Euclidean d–space, d ≥ 3, even for the unit ball K = Bd.