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.
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© 1997 Springer-Verlag Berlin Heidelberg
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Cassels, J.W.S. (1997). Theorems of BLICHFELDT and MINKOWSKI. In: An Introduction to the Geometry of Numbers. Classics in Mathematics, vol 99. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-62035-5_4
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DOI: https://doi.org/10.1007/978-3-642-62035-5_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61788-4
Online ISBN: 978-3-642-62035-5
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