# 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)