Theorems of BLICHFELDT and MINKOWSKI

Abstract

The whole of the geometry of numbers may be said to have sprung from MINKOWSKI’S convex body theorem. In its crudest sense this says that if a point set L in n-dimensional euclidean space is symmetric about the origin (i.e. contains — x when it contains x) and convex [i.e. contains the whole line-segment λx + (1 – λ)y (0 ≦ λ ≦ 1)

when it contains x andy] and has volume V>2ⁿ, then it contains an integral point u other than the origin. In this way we have a link between the “geometrical” properties of a set — convexity, symmetry and volume — and an “arithmetical” property, namely the existence of an integral point in L. Another form of the same theorem, which is more general only in appearance, states that if Λ is a lattice of determinant d(Λ) and L is convex and symmetric about the origin, as before, then L contains a point of Λ other than the origin, provided that the volume V of L is greater than 2ⁿ d(Λ). In § 2 we shall prove MINKOWSKI’S theorem and some refinements. We shall not follow MINKOWSKI’S own proof but deduce his theorem from one of BLICHFELDT, which has important applications of its own and which is intuitively practically obvious: if a point set ℛ has volume strictly greater than d(Λ) then it contains two distinct points x₁ and x ₂ whose difference x ₁ —x ₂ belongs to Λ.