Given a positive integer $k$ and a matrix $X\in \mathbb{R}^{m\times n}$. A truncated Frobenius norm of a matrix $X$ is defined by $$\Vert X \Vert_{k,F} = \sqrt{\sum_{i=k+1}^{m} \sigma_i^2(X)},$$ where $\sigma_i(X)$ are singular values of $X$. I wonder that this norm is convex or not?
For the case Frobenius norm $\Vert \cdot \Vert_F$, I can prove it is convex by showing it is a unitarily invariant norm (since $\Vert \cdot \Vert_2$ is gauge function). But for the case truncated Frobenius norm, I can't use the same technique because $\Vert \cdot \Vert_{k,2}$ is not a gauge function.