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Difference between revisions of "Minkowski theorem"

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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|geometry of numbers]] as a separate division of number theory. It was established by H. Minkowski in 1896 (see [[#References|[1]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m0640901.png" /> be a closed convex body, symmetric with respect to the origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m0640902.png" /> and having volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m0640903.png" />. Then every point lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m0640904.png" /> of determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m0640905.png" /> for which
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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|geometry of numbers]] as a separate division of number theory. It was established by H. Minkowski in 1896 (see [[#References|[1]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m0640901.png" /> be a closed convex body, symmetric with respect to the origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m0640902.png" /> and having volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m0640903.png" />. Then every [[Lattice of points|point lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m0640904.png" /> of determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m0640905.png" /> for which
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m0640906.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m0640906.png" /></td> </tr></table>
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====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Gruber,  C.G. Lekkerkerker,  "Geometry of numbers" , North-Holland  (1987)  pp. Sect. (iv)  (Updated reprint)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Erdös,  P.M. Gruber,  J. Hammer,  "Lattice points" , Longman  (1989)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Gruber,  C.G. Lekkerkerker,  "Geometry of numbers" , North-Holland  (1987)  pp. Sect. (iv)  (Updated reprint)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Erdös,  P.M. Gruber,  J. Hammer,  "Lattice points" , Longman  (1989)</TD></TR></table>
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[[Category:Number theory]]

Revision as of 19:14, 15 November 2014

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)
How to Cite This Entry:
Minkowski theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minkowski_theorem&oldid=34527
This article was adapted from an original article by A.V. Malyshev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article