Gruber P. Convex and Discrete Geometry

24 The Minkowski–Hlawka Theorem
24 The Minkowski–Hlawka Theorem 385
Given an o-symmetric convex body C, Minkowski’s fundamental theorem says that
any lattice L with sufficiently small determinant contains a point of C other than o.
A theorem of Hlawka [509] which verifies a conjecture of Minkowski that the latter
stated in slightly different forms at various places, for example in [733], yields the
following counterpart of this statement: There is a lattice, the determinant of which
is not too large, which contains no point of C other than o.
The Minkowski–Hlawka theorem has attracted interest ever since it was proved
by Hlawka in 1944. In the first decades, emphasis was on alternative proofs, refinements
and generalizations. We mention Siegel, Rogers, Macbeath, Cassels and
Schmidt. More recently, its relations to error correcting codes have been studied
by Sloane and Rush amongst others. A Minkowski–Hlawka theorem, in the adelic
setting, is due to Thunder [998].
In this section we present a basic version of the Minkowski–Hlawka theorem and
state a beautiful generalization of it, the mean value theorem of Siegel. Applications
to lattice packing will be given in Theorem 30.4.
For further pertinent results and numerous references, see [447].
24.1 The Minkowski–Hlawka Theorem
In the following we prove a classical version of the Minkowski–Hlawka theorem.
A Version of the Minkowski–Hlawka Theorem
Theorem 24.1. Let J be a Jordan measurable set in E d with V (J)