Covering by star domains

Abstract

If S is a symmetrical convex domain in R₂ then it is known that the best general covering density θ (S) of S is equal to its best lattice covering density θ_L(S). Examples of symmetrical as well as non- symmetrical star domains S in R₂ are given for which θ (S) < θ _L(S). For cylinders e = S × [-½ , ½] based on symmetrical convex domains S in R₂ one has θ_L(S) = θ_L(C). It is shown that this result need not hold if S is a star domain.