The vector-valued functions associated with circular cones

The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation, which includes second-order cone as a special case when the rotation angle is 45 degrees. Let Lθ denote the circular cone in Rn. For a function f from R to R, one can define a corresponding vector-valued function fLθ on Rn by applying f to the spectral values of the spectral decomposition of x Rn with respect to Lθ. In this paper, we study properties that this vector-valued function inherits from f, including Hölder continuity, B-subdifferentiability, ρ-order semismoothness, and positive homogeneity. These results will play crucial role in designing solution methods for optimization problem involved in circular cone constraints.