Minkowski theorem

Minkowski's theorem on convex bodies is the most important theorem in the geometry of numbers, and is the basis for the existence of the geometry of numbers as a separate division of number theory. It was established by H. Minkowski in 1896 (see [1]). Let be a closed convex body, symmetric with respect to the origin and having volume . Then every point lattice of determinant for which

has a point in distinct from .

An equivalent formulation of Minkowski's theorem is:

where is the critical determinant of the body (see Geometry of numbers). A generalization of Minkowski's theorem to non-convex bodies is Blichfeldt's theorem (see Geometry of numbers). The theorems of Minkowski and Blichfeldt enable one to estimate from above the arithmetic minima of distance functions.

References

[1]	H. Minkowski, "Geometrie der Zahlen" , Chelsea, reprint (1953)

Comments

A refinement of Minkowski's theorem employing Fourier series was given by C.L. Siegel. A different refinement is Minkowski's theorem on successive minima (see Geometry of numbers). These refinements have applications in algebraic number theory and in Diophantine approximation. For a collection of other conditions which guarantee the existence of lattice points in a convex body see .

Minkowski's theorem on linear forms: The system of inequalities

where are real numbers, has an integer solution if . This was established by H. Minkowski in 1896 (see [1]). Minkowski's theorem on linear forms is a corollary of the more general theorem of Minkowski on a convex body (see part 1).

References

[1]	H. Minkowski, "Geometrie der Zahlen" , Chelsea, reprint (1953)
[2]	H. Minkowski, "Diophantische Approximationen" , Chelsea, reprint (1957)
[3]	J.W.S. Cassels, "An introduction to the geometry of numbers" , Springer (1972)

E.I. Kovalevskaya

Comments

The problem when the first inequality in Minkowski's theorem on linear forms can be replaced by strict inequality was solved by G. Hajós.

References

[a1]	P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint)
[a2]	P. Erdös, P.M. Gruber, J. Hammer, "Lattice points" , Longman (1989)