Packing minima and lattice points in convex bodies

Motivated by long-standing conjectures on the discretization of classical inequalities in the geometry of numbers, we investigate a new set of parameters, which we call packing minima, associated to a convex body [math] and a lattice [math] . These numbers interpolate between the successive minima of [math] and the inverse of the successive minima of the polar body of [math] and can be understood as packing counterparts to the covering minima of Kannan & Lovász (1988).¶ As our main results, we prove sharp inequalities that relate the volume and the number of lattice points in [math] to the sequence of packing minima. Moreover, we extend classical transference bounds and discuss a natural class of examples in detail.