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Theorems of BLICHFELDT and MINKOWSKI

  • J. W. S. Cassels
Chapter
Part of the Classics in Mathematics book series (volume 99)

Abstract

The whole of the geometry of numbers may be said to have sprung from MINKOWSKIS 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>2 n , 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 2 n 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 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.

Keywords

Integral Vector Critical Lattice Star Body Ternary Quadratic Form Admissible Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • J. W. S. Cassels
    • 1
  1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK

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