Prox-Regularity of Spectral Functions and Spectral Sets

Important properties such as differentiability and convexity of symmetric functions in $\mathbb{R}^{n}$ can be transferred to the corresponding spectral functions and vice-versa. Continuing to built on this line of research, we hereby prove that a spectral function $F\colon {\bf S}^n \rightarrow \mathbb{R\cup \{+\infty \}}$ is prox-regular if and only if the underlying symmetric function $f\colon\mathbb{R}^{n}\rightarrow \mathbb{R\cup \{+\infty \}}$ is prox-regular. Relevant properties of symmetric sets are also discussed.