$\{A_1 \ldots A_K \}$ is a set of matrices in $\mathbb{R}^{m \times n}$. Let $f (A_1,\ldots,A_K)$ be a function of the singular values of all matrices. For e.g., $f$ is just summation of singular values of all matrices. But potentially I would like to find the class of such functions which are jointly convex in $A_1, \ldots, A_K$. Can necessary and sufficient conditions be established for a function to belong to such a class?