Star body

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.

With every star body $ \mathfrak S $ with respect to the origin $ O $ one can associate, in one-to-one fashion, a 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

[1]	J.W.S. Cassels, "An introduction to the geometry of numbers" , Springer (1972)

Comments

Star bodies play an important role in the 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), $ 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).

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)