VIII The Geometry of Numbers....................................
It was shown by Hermite (1850) that if
f(x)=xtAxis a positive definite quadratic form innreal variables, then there exists a vectorxwith
integercoordinates, not all zero, such that
f(x)≤cn(detA)^1 /n,wherecnis a positive constant depending only onn. Minkowski (1891) found a new
and more geometric proof of Hermite’s result, which gave a much smaller value for the
constantcn. Soon afterwards (1893) he noticed that his proof was valid not only for an
n-dimensional ellipsoidf(x)≤const., but for any convex body which was symmetric
about the origin. This led him to a large body of results, to which he gave the somewhat
paradoxical name ‘geometry of numbers’. It seems fair to say that Minkowski was the
first to realize the importance of convexity for mathematics, and it was in his lattice
point theorem that he first encountered it.
1 Minkowski’s Lattice Point Theorem
AsetC⊆Rnis said to beconvexifx 1 ,x 2 ∈Cimpliesθx 1 +( 1 −θ)x 2 ∈Cfor
0 <θ<1. Geometrically, this means that whenever two points belong to the set the
whole line segment joining them is also contained in the set.
Theindicator functionor ‘characteristic function’ of a setS⊆Rnis defined by
χ(x)=1 or 0 according asx ∈Sorx ∈/S. If the indicator function is Lebesgue
integrable, then the setSis said to havevolume
λ(S)=∫
Rnχ(x)dx.The indicator function of a convex setCis actually Riemann integrable. It is easily
seen that if a convex setCis not contained in a hyperplane ofRn, then itsinterior
intC(see§4 of Chapter I) is not empty. It follows thatλ(C)=0 if and only ifCis
W.A. Coppel, Number Theory: An Introduction to Mathematics, Universitext,
DOI: 10.1007/978-0-387-89486-7_8, © Springer Science + Business Media, LLC 2009
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