Another transference theorem of the geometry of numbers

Abstract

In a recent note (1) I sharpened existing theorems, which state, roughly speaking, that unless a symmetric convex body K has an excessively small homogeneous minimum, it must have a reasonably small inhomogeneous minimum; in other words, if the translates of K centred at points of the integer lattice form an efficient packing, then by expanding them we may derive an efficient covering. In this note I will prove the converse, that a bad packing leads to a bad covering; the result is a much less useful one, as the worst case occurs when we foolishly try to pack rather spiky bodies point to point. A result of this kind has previously been proved by Mahler (4), but his result is less precise from our present point of view; a full account is given by Cassels(3).