Number Theory pp 327-362 | Cite as

The Geometry of Numbers

  • W. A. Coppel
Part of the Universitext book series (UTX)


Minkowski (1891) found a new and more geometric proof of Hermite’s result, which gave a much smaller value for the constant c n . Soon afterwards (1893) he noticed that his proof was valid not only for an n-dimensional ellipsoid f (x) ≤ const., but for any convex body which was symmetric about the origin. This led him to a large body of results, to which he gave the somewhat paradoxical name ‘geometry of numbers’. It seems fair to say that Minkowski was the first to realize the importance of convexity for mathematics, and it was in his lattice point theorem that he first encountered it.


Root Lattice Convex Body Voronoi Cell Nonempty Interior Lattice Packing 
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Additional References

  1. F. Pfender and G. Ziegler, Kissing numbers, sphere packings and some unexpected proofs, Notices Amer. Math. Soc. 51 (2004), 873-883. [The Leech lattice is indeed the densest lattice in <$>\mathbb{R}^{24}<$>.]MATHMathSciNetGoogle Scholar
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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • W. A. Coppel
    • 1
  1. 1.GriffithAustralia

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