Covering Minima and Lattice Point Free Convex Bodies

Abstract:

The covering radius of a convex body K (with respect to a lattice L) is the least factor by which the body needs to be blown up so that its translates by lattice vectors cover the whole space. The covering radius and related quantities have been studied extensively in the Geometry of Numbers (mainly for
convex bodies symmetric about the origin). In this paper, we define and study the "covering minima" of a general convex body. The covering radius will be one of these minima; the "lattice width" of the body will be the reciprocal of another. We derive various inequalities relating these minima. These imply
bounds on the width of lattice point free convex bodies. We prove that every lattice-point-free body has a projection whose volume is not much larger than the determinant of the projected lattice.