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In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by Hermann Minkowski.

The geometry of numbers has a close relationship with other fields of mathematics, especially functional analysis and Diophantine approximation, the problem of finding rational numbers that approximate an irrational quantity.^[1]

Minkowski's results

Minkowski's first theorem states that a convex body with sufficient volume that is symmetric around the origin contains a nonzero vector with integer coordinates. Minkwoski's second theorem states that a convex body with sufficient volume (and with the same symmetry requirement) contains a basis of vectors, each having integer coordinates.^[2]

Later research in the geometry of numbers

In 1930-1960 research on the geometry of numbers was conducted by many number theorists (including Louis Mordell, Harold Davenport and Carl Ludwig Siegel). Contemporary research has been done by many authors, some of whose works are listed in the selective bibliography. In recent years, Lenstra, Brion, and Barvinok have developed combinatorial theories that enumerate the lattice points in some convex bodies.^[3]

Subspace theorem of W. M. Schmidt

In the geometry of numbers, the subspace theorem was obtained by Wolfgang M. Schmidt in 1972.^[4] It states that if L₁,...,L_n are linearly independent linear forms in n variables with algebraic coefficients and if ε>0 is any given real number, then the non-zero integer points x with

lie in a finite number of proper subspaces of Qⁿ.

Influence on functional analysis

Minkowski's geometry of numbers had a profound influence on functional analysis. Minkowski proved that symmetric convex bodies induce norms in finite-dimensional vector spaces. Minkwoski's theorem was generalized to topological vector spaces by Kolmogorov, whose theorem states that the symmetric convex sets that are closed and bounded generate the topology of a Banach space.^[5]

Researchers continue to study generalizations to star-shaped sets and other non-convex sets.^[6]

References

1. ^ Schmidt's books. Grötschel et alia, Lovász et alia, Lovász.
2. ^ Minkowski, Hancock, Cassels, Siegel, Lekkerkererker, Hlawka et alia. See the extensions by Bombieri and Vaaler and by Schmidt.
3. ^ Grötschel et alia, Lovász et alia, Lovász, and Beck and Robins.
4. ^ Schmidt, Wolfgang M. Norm form equations. Ann. of Math. (2) 96 (1972), pp. 526-551. See also Schmidt's books; compare Bombieri and Vaaler and also Bombieri and Gubler.
5. ^ For Kolmogorov's normability theorem, see Walter Rudin's Functional Analysis. For more results, see Schneider, and Thompson and see Kalton et alia.
6. ^ Kalton et alia. Gardner

Bibliography

• Matthias Beck, Sinai Robins. Computing the continuous discretely: Integer-point enumeration in polyhedra, Undergraduate texts in mathematics, Springer, 2007.
• Enrico Bombieri; Vaaler, J. (Feb 1983). "On Siegel's lemma". Inventiones Mathematicae 73 (1): 11–32. doi:10.1007/BF01393823. http://www.springerlink.com/content/k55042224131lp42. 
• Enrico Bombieri and Walter Gubler (2006). Heights in Diophantine Geometry. Cambridge U. P.. 
• J. W. S. Cassels. An Introduction to the Geometry of Numbers. Springer Classics in Mathematics, Springer-Verlag 1997 (reprint of 1959 and 1971 Springer-Verlag editions).
• John Horton Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, NY, 3rd ed., 1998.
• R. J. Gardner, Geometric tomography, Cambridge University Press, New York, 1995. Second edition: 2006.
• P. M. Gruber, Convex and discrete geometry, Springer-Verlag, New York, 2007.
• P. M. Gruber, J. M. Wills (editors), Handbook of convex geometry. Vol. A. B, North-Holland, Amsterdam, 1993.
• M. Grötschel, L. Lovász, A. Schrijver: Geometric Algorithms and Combinatorial Optimization, Springer, 1988
• Hancock, Harris (1939). Development of the Minkowski Geometry of Numbers. Macmillan.  (Republished in 1964 by Dover.)
• Hazewinkel, Michiel, ed. (2001), "Geometry of numbers", Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/G/g044350.htm 
• Edmund Hlawka, Johannes Schoißengeier, Rudolf Taschner. Geometric and Analytic Number Theory. Universitext. Springer-Verlag, 1991.
• Kalton, Nigel J.; Peck, N. Tenney; Roberts, James W. (1984), An F-space sampler, London Mathematical Society Lecture Note Series, 89, Cambridge: Cambridge University Press, pp. xii+240, ISBN 0-521-27585-7  MR0808777
• C. G. Lekkerkererker. Geometry of Numbers. Wolters-Noordhoff, North Holland, Wiley. 1969.
• Lenstra, A. K.; Lenstra, H. W., Jr.; Lovász, L. (1982). "Factoring polynomials with rational coefficients". Mathematische Annalen 261 (4): 515–534. doi:10.1007/BF01457454. MR0682664. http://hdl.handle.net/1887/3810. 
• L. Lovász: An Algorithmic Theory of Numbers, Graphs, and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986
• Wolfgang M. Schmidt. Diophantine approximation. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])
• Wolfgang M. Schmidt.Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, Springer Verlag 2000.
• Siegel, Carl Ludwig (1989). Lectures on the Geometry of Numbers. Springer-Verlag. 
• Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, 1993.
• Anthony C. Thompson, Minkowski geometry, Cambridge University Press, Cambridge, 1996.

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