Advertisement

Lattices and Minkowski’s Theorem

  • Jiří Matoušek
Part of the Graduate Texts in Mathematics book series (GTM, volume 212)

Abstract

This chapter is a quick excursion into the geometry of numbers, a field where number-theoretic results are proved by geometric arguments, often using properties of convex bodies in R d . We formulate the simple but beautiful theorem of Minkowski on the existence of a nonzero lattice point in every symmetric convex body of sufficiently large volume. We derive several consequences, concluding with a geometric proof of the famous theorem of Lagrange claiming that every natural number can be written as the sum of at most 4 squares.

Keywords

Convex Body Discrete Subgroup Integer Point Symmetric Convex Integer Coefficient 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag New York, Inc. 2002

Authors and Affiliations

  • Jiří Matoušek
    • 1
  1. 1.Department of Applied MathematicsCharles UniversityPraha 1Czech Republic

Personalised recommendations