Suppose that is a lattice in -dimensional Euclidean space and is a convex centrally symmetric body.
Minkowski's theorem, sometimes called Minkowski's first theorem, states that if , then contains a nonzero vector in .
The successive minimum is defined to be the inf of the numbers such that contains linearly independent vectors of .
Minkowski's theorem on successive minima, sometimes called Minkowski's second theorem, is a strengthening of his first theorem and states that[4]
In 1930–1960 research on the geometry of numbers was conducted by many number theorists (including Louis Mordell, Harold Davenport and Carl Ludwig Siegel). In recent years, Lenstra, Brion, and Barvinok have developed combinatorial theories that enumerate the lattice points in some convex bodies.[5]
In the geometry of numbers, the subspace theorem was obtained by Wolfgang M. Schmidt in 1972.[6] It states that if n is a positive integer, and L1,...,Ln are linearly independentlinearforms in n variables with algebraic coefficients and if ε>0 is any given real number, then the non-zero integer points x in n coordinates with
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. Minkowski'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.[7]
Schmidt, Wolfgang M. Norm form equations. Ann. Math. (2) 96 (1972), pp. 526–551.
See also Schmidt's books; compare Bombieri and Vaaler and also Bombieri and Gubler.
Enrico Bombieri & 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).
Hancock, Harris (1939). Development of the Minkowski Geometry of Numbers. Macmillan. (Republished in 1964 by Dover.)
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, ISBN0-521-27585-7, MR0808777
C. G. Lekkerkererker. Geometry of Numbers. Wolters-Noordhoff, North Holland, Wiley. 1969.
Lovász, L.: An Algorithmic Theory of Numbers, Graphs, and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986