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''on the product of inhomogeneous linear forms''

A statement according to which for real linear forms

$$ 
L _ {j} ( \overline{x}\; )  = \ 
a _ {j1} x _ {1} + \dots + a _ {jn} x _ {n} ,\ \ 
1 \leq  j \leq  n,
$$

in  $  n $
variables  $  x _ {1} \dots x _ {n} $, 
with a non-zero determinant  $  \Delta $, 
and any real  $  \alpha _ {1} \dots \alpha _ {n} $, 
there are integers  $  x _ {1} \dots x _ {n} $
such that the inequality

$$ \tag{* }
\prod _ { j= } 1 ^ { n } 
| L _ {j} ( \overline{x}\; ) - \alpha _ {j} |
 \leq   2  ^ {-} n | \Delta |
$$

holds. This hypothesis was proved by H. Minkowski (1918) in case  $  n = 2 $. 
A proof of the hypothesis is known (1982) for  $  n \leq  5 $, 
and (*) has been proved for  $  n > 5 $
under certain additional restrictions (see [[#References|[2]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.W.S. Cassels,   "An introduction to the geometry of numbers" , Springer  (1972)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.F. Skubenko,   "A proof of Minkowski's conjecture on the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064050/m06405011.png" /> linear inhomogeneous forms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064050/m06405012.png" /> variables for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064050/m06405013.png" />" , ''Investigations in number theory'' , ''Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov'' , '''33'''  (1973)  pp. 6–36  (In Russian)</TD></TR></table>

====Comments====
See also [[Geometry of numbers|Geometry of numbers]].

====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>