Nonsmooth Analysis of Lorentz Invariant Functions

A real valued function $g(x,t)$ on ${\mathbb{R}}^n \times {\mathbb{R}}$ is called a Lorentz invariant if $g(x,t)=g(Ux,t)$ for all $n \times n$ orthogonal matrices $U$ and all $(x,t)$ in the domain of $g$. In other words, $g$ is invariant under the linear orthogonal transformations preserving the Lorentz cone: $\{(x,t) \in {\mathbb{R}}^n \times {\mathbb{R}} \,|\, t \ge \|x\| \}$. It is easy to see that every Lorentz invariant function can be decomposed as $g=f \circ \beta$, where $f : {\mathbb{R}}^2 \rightarrow {\mathbb{R}}$ is a symmetric function and $\beta$ is the root map of the hyperbolic polynomial $p(x,t)=t^2-x_1^2-\cdots -x_n^2$. We investigate a variety of important variational and nonsmooth properties of $g$ and characterize them in terms of the symmetric function $f$.