Lattices and Minkowski’s Theorem

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


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.


Convex Body Discrete Subgroup Integer Point Symmetric Convex Integer Coefficient 
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Copyright information

© Springer-Verlag New York, Inc. 2002

Authors and Affiliations

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

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