Gruber P. Convex and Discrete Geometry

366 Geometry of Numbers
22 Minkowski’s First Fundamental Theorem
We cite Cassels [195], prologue:
We owe to MINKOWSKI the fertile observation that certain results which can be
made almost intuitive by consideration of figures in n-dimensional Euclidean space
have far-reaching consequences in diverse branches of number theory. For example,
he simplified the theory of units in algebraic number fields and both simplified
and extended the theory of approximation of irrational numbers by rational ones
(Diophantine Approximation). This new branch of number theory, which
MINKOWSKI christened “The Geometry of Numbers”, has developed into an independent
branch of number theory which, indeed, has many applications elsewhere
but which is well worth studying for its own sake.
Minkowski’s fundamental theorem is one among a small number of basic results
of the geometry of numbers alluded to by Cassels. It relates the basic notions of lattices
and convex bodies. The fundamental theorem is simple, almost trivial and, at the
same time, deep. There exist numerous generalizations and arithmetic consequences
of it. The following remark of Hilbert well describes the situation, see Rose [856]:
The art of doing mathematics consists in finding that special case which contains all
the germs of generality.
In this section we present the fundamental theorem together with some of its classical
applications dealing with Diophantine approximation, representation of integers as
sums of squares, and estimates for discriminants of polynomials.
For a wealth of different versions, related results and further applications, see
Cassels [195], Kannan [563], Gruber and Lekkerkerker [447] and Erdös, Gruber and
Hammer [307].
22.1 The First Fundamental Theorem
Let f : E d → R. The arithmetic problem of finding a solution u = (u1,...,ud) of
the inequality
f (x1,...,xd) ≤ 1,
where the ui are integers, not all equal to 0, is equivalent to the geometric problem
of finding a point of the integer lattice Z d different from o and contained in the set
{x : f (x) ≤ 1}.
This led Minkowski to look for conditions which guarantee that a set, in particular a
convex body, contains points of a lattice different from the origin. The result was his
first fundamental theorem.
In this section we present Minkowski’s theorem together with three elegant and
convincing proofs, each based on a different idea.