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The problem about the homogeneous arithmetical minima of positive  $  n $-
ary quadratic forms with real coefficients. It is equivalent to the problem of the densest lattice packing of  $  n $-
dimensional balls of equal radius (see [[Geometry of numbers|Geometry of numbers]]).

Let  $  f = f ( x) $, 
$  x \in \mathbf R  ^ {n} $, 
be a positive quadratic form over  $  \mathbf R $
of determinant  $  d = d ( f  ) =  \mathop{\rm det}  f \neq 0 $; 
and let

$$ 
m ( f  )  =  \inf _ {\begin{array}{c}
x \in \mathbf Z  ^ {n} \\

x \neq 0 
\end{array}
 }
 f ( x)  =  \min _ {\begin{array}{c}
x \in \mathbf Z  ^ {n} \\

{x \neq 0 } 
\end{array}
 }  f ( x)
$$

be its homogeneous arithmetical minimum. The quantity

$$ 
\gamma _ {n}  =  \sup  
\frac{m ( f  ) }{\{ d ( f  ) \}  ^ {1/n} }
  =  \max \ 

\frac{m ( f  ) }{\{ d ( f  ) \}  ^ {1/n} }
 ,
$$

where the supermum or maximum is over all positive quadratic forms  $  f $, 
is called the Hermite constant;  $  \gamma _ {n} = \{ \gamma ( F _ {n} ) \}  ^ {2} $, 
where  $  F _ {n} ( x) = ( x _ {1}  ^ {2} + \dots + x _ {n}  ^ {2} )  ^ {1/2} $
is the radial function corresponding to a ball.

Originally, one understood by the Hermite problem the task of finding or estimating  $  \gamma _ {n} $(
from above and below). The exact values of  $  \gamma _ {n} $
are known only for  $  n \leq  8 $(
see [[#References|[1]]]). For estimates of  $  \gamma _ {n} $, 
see [[#References|[2]]] or [[#References|[1]]].

Subsequently, the term Hermite problem was used for the search for local maxima (boundary or extremal) of  $  m ( f  ) / \{ d ( f  ) \}  ^ {1/n} $
in the space of coefficients and their corresponding forms  $  f $. 
Algorithms are known for finding all classes of boundary forms. In particular, the Voronoi algorithm for perfect forms (see [[#References|[1]]], [[#References|[3]]], [[#References|[4]]]).

The problem was posed by Ch. Hermite in 1850.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) (Updated reprint) {{MR|0893813}} {{ZBL|0611.10017}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> C.A. Rogers, "Packing and covering" , Cambridge Univ. Press (1964) {{MR|0172183}} {{ZBL|0176.51401}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> B.N. Delone, "The Peterburg school of number theory" , Moscow-Leningrad (1947) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E.P. Baranovskii, "Packings, coverings, partitionings and certain other distributions in spaces of constant curvature" ''Progress in Math.'' , '''9''' (1971) pp. 209–253 ''Itogi Nauk. Mat. Algebra Topol. Geom. 1967'' (1969) pp. 189–225</TD></TR></table>