Thin Lattice Coverings

Let G be a compact body of positive volume in R n , star‐shaped with respect to an interior point, taken to be the origin. For subsets Ω of R n , the functionalI L ( G , Ω ) = V o l ( G )  s u p l a t t i c e s Λ s{ d e t Λ : Ω ⊆ Λ + G }represents the minimum density with which Ω can be covered by a lattice ∧ of translates of G . We obtain an upper bound on I L ( G, Z n ). If the attributes of G are supplemented with convexity, write H instead. We also bound abovem ( r , f ′ / f ) = 1 2 π∫ 0 2 πlog + | f ′ ( r e t 0 ) / f ( r e t 0 ) | d θthe classical minimum lattice‐covering density of H . No symmetry conditions are imposed on G and H .