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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
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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 $N$ be a closed convex body, symmetric with respect to the origin $0$ and having volume $V(N)$. Then every [[Lattice of points|point lattice]] $\Lambda$ of determinant $d(\Lambda)$ for which
 +
$$
 +
V(N) \ge 2^n d(\Lambda)
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$$
  
<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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has a point in $N$ distinct from $0$.
 
 
has a point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m0640907.png" /> distinct from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m0640908.png" />.
 
  
 
An equivalent formulation of Minkowski's theorem is:
 
An equivalent formulation of Minkowski's theorem is:
 
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$$
<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/m0640909.png" /></td> </tr></table>
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\Delta(N) \ge 2^{-n} d(\Lambda)
 
+
$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m06409010.png" /> is the critical determinant of the body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m06409011.png" /> (see [[Geometry of numbers|Geometry of numbers]]). A generalization of Minkowski's theorem to non-convex bodies is Blichfeldt's theorem (see [[Geometry of numbers|Geometry of numbers]]). The theorems of Minkowski and Blichfeldt enable one to estimate from above the arithmetic minima of distance functions.
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where $\Delta(N)$ is the critical determinant of the body $N$ (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====
 
====References====
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====Comments====
 
====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|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 .
+
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
 
Minkowski's theorem on linear forms: The system of inequalities
 
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$$
<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/m06409012.png" /></td> </tr></table>
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\left\vert{ \sum a_{1j} x_j }\right\vert \le c_1
 
+
$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m06409013.png" /> are real numbers, has an integer solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m06409014.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064090/m06409015.png" />. This was established by H. Minkowski in 1896 (see [[#References|[1]]]). Minkowski's theorem on linear forms is a corollary of the more general theorem of Minkowski on a convex body (see part 1).
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$$
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\left\vert{ \sum a_{ij} x_j }\right\vert < c_i\ \ \ i=2,\ldots,n
 +
$$
 +
where $a_{i,j}, c_i$ are real numbers, has an integer solution $(x_1,\ldots,x_n) \neq 0$ if $c_1\cdots c_n \ge |\det a_{i,j}|$. This was established by H. Minkowski in 1896 (see [[#References|[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====
 
====References====
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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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 +
{{TEX|done}}
  
 
[[Category:Number theory]]
 
[[Category:Number theory]]

Revision as of 19:25, 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 $N$ be a closed convex body, symmetric with respect to the origin $0$ and having volume $V(N)$. Then every point lattice $\Lambda$ of determinant $d(\Lambda)$ for which $$ V(N) \ge 2^n d(\Lambda) $$

has a point in $N$ distinct from $0$.

An equivalent formulation of Minkowski's theorem is: $$ \Delta(N) \ge 2^{-n} d(\Lambda) $$ where $\Delta(N)$ is the critical determinant of the body $N$ (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 $$ \left\vert{ \sum a_{1j} x_j }\right\vert \le c_1 $$ $$ \left\vert{ \sum a_{ij} x_j }\right\vert < c_i\ \ \ i=2,\ldots,n $$ where $a_{i,j}, c_i$ are real numbers, has an integer solution $(x_1,\ldots,x_n) \neq 0$ if $c_1\cdots c_n \ge |\det a_{i,j}|$. 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=34528
This article was adapted from an original article by A.V. Malyshev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article