Ray function

A real-valued function defined on an -dimensional space and satisfying the following conditions: is continuous, non-negative and homogeneous (that is, for any real number ). A ray function is said to be positive if for all , and symmetric if . A ray function is said to be convex if for any ,

For any ray function there is a constant for which

If is positive, then there is also a constant for which

The set of points satisfying the condition

is a star body. Conversely, for any open star body there is a unique ray function for which

A star body is bounded if and only if its ray function is positive. If is a symmetric function, then is symmetric about the point 0; the converse is also true. A star body is convex if and only if 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.))