A convex function $f$ is $R$-Lipschitz w.r.t. to a norm $\|\cdot\|$ if for all points $a, b$ we have $|f(a)-f(b)| \leq R\|a-b\|$.
For a real symmetric $n\times n$ matrix $A$ with eigenvalues denoted by $\lambda_i$ and real number $p\geq 1$, the $p$-Schatten norm of $A$ is defined as $\|A\|_p = (\sum_{i=1}^{n} |\lambda_i|^p)^{1/p}$.
Does anyone know of examples of functions on real, symmetric $n\times n$ matrices that are Lipschitz w.r.t to the Schatten $p$-norms (even ones that hold for just particular values of $p$?)?