Let $G$ be any set of orthogonal linear transformations of $R^{n}$ onto itself. A function $f$ is said to be symmetric with respect to $G$ if $$ f(A x)=f(x), \quad \forall x, \quad \forall A \in G . $$ The functions on $R^{n}$ which are symmetric with respect to the set of all orthogonal transformations of $R^{n}$ are, of course, those of the form $$ f(x)=g(|x|), $$ where $|\cdot|$ is the Euclidean norm and $g$ is a function on $[0,+\infty)$.
The above discussion is in Convex Analysis, but I don't quite understand the following:
- How the function $g$ is constructed?
- Why $f$ has such a form?