Gruber P. Convex and Discrete Geometry

Proof. Consider the linear forms
l1(u) = u1 + t1u2 +···+t d−1
1
ud
l2(u) = u1 + t2u2 +···+t d−1
2
ud
................................
ld(u) = u1 + tdu2 +···+t d−1
d ud,
23 Successive Minima 375
where t1,...,td are the roots of p. Foru ∈ Z d \{o}, the linear forms li(u),
i = 1,...,d, all are different from 0, for, if li(u) = 0, then ti is a root of the
polynomial li(u) of degree at most d − 1inti with integer coefficients. This contradicts
the irreducibility of p. Therefore l1(u) ···ld(u) �= 0 and, since this product is
a symmetric polynomial in t1,...,td with integer coefficients, it must be an integer.
Thus
|l1(u) ···ld(u)| ≥1foru ∈ Z d \{o}.
The lattice L ={l = (l1(u),...,ld(u)) : u ∈ Z d } is thus admissible for the star set
{x :|x1 ···xd| ≤1}.
By the inequality of the geometric and arithmetic mean, this star set contains the
o-symmetric cross-polytope
O = � x : 1 �
|x1|+···+|xd|
d
� ≤ 1 � ,
of volume 2 d d d /d !. Thus L is also admissible for O. The fundamental theorem then
shows that d(L) ≥ d d /d !. Noting that D = d(L) 2 , the proof is complete. ⊓⊔
23 Successive Minima
Successive minima of star bodies or convex bodies with respect to lattices were first
defined and investigated by Minkowski in the context of the geometry of numbers.
Minkowski put them to use in algebraic number theory. A hundred years later, successive
minima still play a role in the geometry of numbers and in algebraic and
transcendental number theory. See, e.g. Bertrand [102], Chen [204] and Matveev
[697], but they are also important in Diophantine Approximation, see, e.g. Schmidt
[896], and in computational geometry, compare Lagarias, Lenstra and Schnorr [626],
Schnorr [913] and Blömer [134]. There is a surprising link to Nevanlinna’s value distribution
theory, a branch of complex analysis, see Wong [1028] and Hyuga [534].
Relations to lattice polytopes and roots of Ehrhart polynomials were studied by
Stanley, Henk, Schürmann and Wills, see the references in Sect. 19.1 and the survey
of Henk and Wills [493].
In this section, we first present Minkowski’s theorem on successive minima and
prove a result of Mahler relating successive minima of a convex body with respect
to a lattice and successive minima of the polar body with respect to the polar lattice.
Then Jarník’s transference theorem is proved. It connects lattice packing and
lattice covering of a given convex body. Finally, we give a result of Perron and
Khintchine on Diophantine approximation.