Covering Minima and Lattice-Point-Free Convex Bodies

The radius of a convex body K (with respect to a L) is the least factor by which the body needs to be blown up so that its translates by vectors cover the whole space. The 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 radius will be one of these minima; the lattice 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.