View source for 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|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>

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:

<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>

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Minkowski,   "Geometrie der Zahlen" , Chelsea, reprint  (1953)</TD></TR></table>



====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 .

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

<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>

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).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Minkowski,   "Geometrie der Zahlen" , Chelsea, reprint  (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Minkowski,   "Diophantische Approximationen" , Chelsea, reprint  (1957)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.W.S. Cassels,   "An introduction to the geometry of numbers" , Springer  (1972)</TD></TR></table>

''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====
<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>

[[Category:Number theory]]