340 VIII The Geometry of Numbers
It follows from Proposition 12 that∆(K)≥ 2 −nλ(K). A conjectured sharpening
of Minkowski’s theorem on successive minima, which has been proved by Minkowski
(1896) himself forn=2andforn-dimensional ellipsoids, and by Woods (1956) for
n=3, claims that
μ 1 μ 2 ···μn∆(K)≤d(Λ).The successive minima of a convex body are connected with those of its dual body.
IfKis a compact symmetric convex subset ofRnwith nonempty interior, then itsdual
K∗={y∈Rn:ytx≤1forallx∈K}has the same properties, andKis the dual ofK∗. Mahler (1939) showed that the
successive minima of the dual bodyK∗with respect to the dual latticeΛ∗are related
to the successive minima ofKwith respect toΛby the inequalities
1 ≤μi(K,Λ)μn−i+ 1 (K∗,Λ∗)(i= 1 ,...,n),and hence, by applying Minkowski’s theorem on successive minima also toK∗and
Λ∗, he obtained inequalities in the opposite direction:
μi(K,Λ)μn−i+ 1 (K∗,Λ∗)≤ 4 n/λ(K)λ(K∗)(i= 1 ,...,n).By further proving thatλ(K)λ(K∗)≥ 4 n(n!)−^2 , he deduced that
μi(K,Λ)μn−i+ 1 (K∗,Λ∗)≤(n!)^2 (i= 1 ,...,n).Dramatic improvements of these bounds have recently been obtained. Banaszczyk
(1996), with the aid of techniques from harmonic analysis, has shown that there is
a numerical constantC>0suchthat,foralln≥1andalli∈{ 1 ,...,n),
μi(K,Λ)μn−i+ 1 (K∗,Λ∗)≤Cn( 1 +logn).He had shown already (1993) that ifK=B 1 is then-dimensional closed unit ball,
which is self-dual, then for alln≥1andalli∈{ 1 ,...,n),
μi(B 1 ,Λ)μn−i+ 1 (B 1 ,Λ∗)≤n.This result is close to being best possible, since there exists a numerical constant
C′>0 and self-dual latticesΛn⊆Rnsuch that
μ 1 (B 1 ,Λn)μn(B 1 ,Λn)≥μ 1 (B 1 ,Λn)^2 ≥C′n.Two other applications of Minkowski’s theorem on successive minima will be men-
tioned here. The first is a sharp form, due to Bombieri and Vaaler (1983), of ‘Siegel’s
lemma’. In his investigations on transcendental numbers Siegel (1929) used Dirichlet’s
pigeonhole principle to prove that ifA=(αjk)is anm×nmatrix of integers, where