Nonsmooth Analysis of Lorentz Invariant Functions

A real valued function g(x, t) on Rn×R is called Lorentz invariant if g(x, t) = g(Ux, t) for all n×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) ∈ Rn × R | t ≥ ‖x‖}. It is easy to see that every Lorentz invariant function can be decomposed as g = f ◦ β, where f : R2 → R is a symmetric function and β is the root map of the hyperbolic polynomial p(x, t) = t−x1−· · ·−xn. We investigate variety of important variational and non-smooth properties of g and characterize them in terms of the symmetric function f .