Packing minima and lattice points in convex bodies

Abstract

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 $K$ and a lattice $\mathrm{\Lambda }$. These numbers interpolate between the successive minima of $K$ and the inverse of the successive minima of the polar body of $K$ 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 $K$ to the sequence of packing minima. Moreover, we extend classical transference bounds and discuss a natural class of examples in detail.