328 VIII The Geometry of Numbers
contained in a hyperplane, and 0<λ(C)<∞if and only ifCis bounded and is not
contained in a hyperplane.
AsetS⊆Rnis said to besymmetric(with respect to the origin) ifx∈Simplies
−x∈S. Evidently any (nonempty) symmetric convex set contains the origin.
A pointx=(ξ 1 ,...,ξn)∈Rnwhose coordinatesξ 1 ,...,ξnare all integers will
be called alattice point. Thus the set of all lattice points inRnisZn.
These definitions are the ingredients for Minkowski’slattice point theorem:
Theorem 1Let C be a symmetric convex set inRn.Ifλ(C)> 2 n, or if C is compact
andλ(C)= 2 n, then C contains a nonzero point ofZn.
The proof of Theorem 1 will be deferred to§3. Here we illustrate the utility of the
result by giving several applications, all of which go back to Minkowski himself.
Proposition 2If A is an n×n positive definite real symmetric matrix, then there exists
a nonzero point x∈Znsuch that
xtAx≤cn(detA)^1 /n,where cn=( 4 /π){(n/ 2 )!}^2 /n.
Proof For anyρ>0 the ellipsoidxtAx ≤ ρis a compact symmetric convex
set. By puttingA=TtT, for some nonsingular matrixT, it may be seen that the
volume of this set isκnρn/^2 (detA)−^1 /^2 ,whereκnis the volume of then-dimensional
unit ball. It follows from Theorem 1 that theellipsoid contains a nonzero lattice point
ifκnρn/^2 (detA)−^1 /^2 = 2 n. But, as we will see in§4 of Chapter IX,κn=πn/^2 /(n/ 2 )!,
wherex!=Γ(x+ 1 ). This gives the valuecnforρ.
It follows from Stirling’s formula (Chapter IX,§4) thatcn∼ 2 n/πeforn→∞.
Hermite had proved Proposition 2 withcn=( 4 / 3 )(n−^1 )/^2. Hermite’s value is smaller
than Minkowski’s forn≤8, but much larger for largen.
As a second application of Theorem 1 we prove Minkowski’slinear forms theorem:
Proposition 3Let A be an n×n real matrix with determinant± 1. Then there exists
a nonzero point x∈Znsuch that Ax=y=(ηk)satisfies
|η 1 |≤ 1 , |ηk|< 1 for 1 <k≤n.Proof For any positive integerm,letCmbe the set of allx∈Rnsuch thatAx∈Dm,
where
Dm={y=(ηk)∈Rn:|η 1 |≤ 1 + 1 /m,|ηk|<1for2≤k≤n}.ThenCmis a symmetric convex set, sinceAis linear andDmis symmetric and convex.
Moreoverλ(Cm)= 2 n( 1 + 1 /m),sinceλ(Dm)= 2 n( 1 + 1 /m)andAis volume-
preserving. Therefore, by Theorem 1,Cmcontains a lattice pointxm = O.Since
Cm⊂C 1 for allm>1 and the number of lattice points inC 1 is finite, there exist only
finitely many distinct pointsxm. Thus there exists a lattice pointx=Owhich belongs
toCmfor infinitely manym. Evidentlyxhas the required properties.