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''with respect to a point  $  O $, 
star-like body''

An open set  $  \mathfrak S $
in  $  n $-
dimensional Euclidean space  $  \mathbf R  ^ {n} $
which has the ray property (relative to  $  O $): 
If  $  a \in \overline{\mathfrak S}\; $, 
where  $  \overline{\mathfrak S}\; $
is the closure of  $  \mathfrak S $, 
then the entire segment  $  [ O , a ) $(
where  $  O \in [ O , a ) $, 
$  a \notin [ O , a ) $) 
lies in  $  \mathfrak S $. 
A star body  $  \mathfrak S $
with centre  $  O $
may be characterized as follows:  $  O $
is an interior point of  $  \mathfrak S $; 
every ray emanating from  $  O $
lies either entirely in  $  \mathfrak S $
or contains a point  $  a $
such that the ray segment  $  [ O , a ) $
lies in  $  \mathfrak S $, 
but the ray segment  $  ( a, + \infty ) $
lies outside  $  \mathfrak S $. 
This definition is equivalent to the first one, up to points on the boundary of  $  \mathfrak S $. 
A star body is a particular case of a star set with respect to  $  O $, 
a set with the generalized ray property relative to  $  O $: 
If  $  a \in \mathfrak S $, 
then the entire segment  $  [ O , a ] $
lies in  $  \mathfrak S $. 
A particular case of a star body is a [[Convex body|convex body]].

With every star body  $  \mathfrak S $
with respect to the origin  $  O $
one can associate, in one-to-one fashion, a [[Ray function|ray function]]  $  F ( x ) = F _ {\mathfrak S} ( x ) $
such that  $  \mathfrak S $
is the set of points  $  x \in \mathbf R  ^ {n} $
with  $  F ( x ) < 1 $.

The correspondence is defined by the formula

$$ 
F ( x )  =  \inf _ {\begin{array}{c}
{tx \in \mathfrak S } \\
 {t > 0 } 
\end{array}
 }
 
\frac{1}{t}
 .
$$

With this notation a star body  $  \mathfrak S $
is bounded if and only if  $  F ( x ) $
is a positive ray function; it is convex if and only if  $  F ( x ) $
is a convex ray function.

====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></table>

====Comments====
Star bodies play an important role in the [[Geometry of numbers|geometry of numbers]], e.g. the Minkowski–Hlawka theorem.

A set  $  S $
in  $  \mathbf R  ^ {n} $
is centrally symmetric if  $  x \in S $
implies  $  - x \in S $.

The Minkowski–Hlawka theorem says that  $  V ( \mathfrak S ) \geq  2 \zeta ( n) \Delta ( \mathfrak S ) $
for a centrally-symmetric star body  $  \mathfrak S $. 
Here,  $  \Delta ( \mathfrak S ) $
is the critical determinant of  $  \mathfrak S $(
cf. [[Geometry of numbers|Geometry of numbers]]),  $  V( \mathfrak S ) $
is the volume of  $  \mathfrak S $
and  $  \zeta ( n) = 1+ 2  ^ {-} n + 3  ^ {-} n + \dots $. 
This is an inequality in the opposite direction of the Minkowski convex body theorem (cf. [[Minkowski theorem|Minkowski theorem]]).

====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>