Ray function

A real-valued function $ F ( x) $ defined on an $ n $- dimensional space $ \mathbf R ^ {n} $ and satisfying the following conditions: $ F ( x) $ is continuous, non-negative and homogeneous (that is, $ F ( \tau x) = \tau F ( x) $ for any real number $ \tau \geq 0 $). A ray function $ F ( x) $ is said to be positive if $ F ( x) > 0 $ for all $ x \neq 0 $, and symmetric if $ F ( - x ) = F ( x) $. A ray function is said to be convex if for any $ x , y \in \mathbf R ^ {n} $,

$$ F ( x + y ) \leq F ( x) + F ( y) . $$

For any ray function $ F ( x) $ there is a constant $ c = c _ {F} $ for which

$$ F ( x) \leq c | x | ,\ x \in \mathbf R ^ {n} . $$

If $ F ( x) $ is positive, then there is also a constant $ \widetilde{c} = \widetilde{c} _ {F} > 0 $ for which

$$ F ( x) \geq \widetilde{c} | x | ,\ x \in \mathbf R ^ {n} . $$

The set $ \mathfrak C $ of points $ x \in \mathbf R ^ {n} $ satisfying the condition

$$ F ( x) < 1 $$

is a star body. Conversely, for any open star body $ \mathfrak C $ there is a unique ray function $ F _ {\mathfrak C } ( x) $ for which

$$ \mathfrak C = \{ {x } : {F _ {\mathfrak C } ( x) < 1 } \} . $$

A star body $ \mathfrak C _ {F} $ is bounded if and only if its ray function $ F ( x) $ is positive. If $ F ( x) $ is a symmetric function, then $ \mathfrak C _ {F} $ is symmetric about the point 0; the converse is also true. A star body 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 (1959)

Comments

Star bodies are usually defined as closed ray sets. A ray function is more commonly called a distance function.

References

[a1]	P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint)
[a2]	E. Hlawka, "Das inhomogene Problem in der Geometrie der Zahlen" , Proc. Internat. Congress Mathematicians (Amsterdam, 1954) , 3 , Noordhoff (1954) pp. 20–27 ((Also: Selecta, Springer 1990, 178–185.))